geometrytiling
According to Wikipedia and one of its sources (archived), the terms are defined as follows:
These seem completely equivalent to me.
So, you can literally write a book about this stuff (exhibit a) and so I will try my best to keep it short and sweet (as much as possible at least). I'll also not mention anything about the physical interpretation of aperiodic patterns and tilings which mathematical and solid state physicists are interested in - namely quasicrystals.
You mentioned that periodic tilings can be characterised by a group action of the space that they tile under which the tiling is invariant - let's stick to euclidean $2$-space, $\mathbb{R}^2$ for this post. You can do this in a few ways, you could either look at just the subgroup of the translation group on $\mathbb{R}^2$ under which the tiling $\mathcal{T}$ is invariant, in which case a cocompact periodic tiling has invariant subgroup isomorphic to $\mathbb{Z}^2$, or you can consider the subgroup of the full euclidean group on $\mathbb{R}^2$ under which $\mathcal{T}$ is invariant, including rotations - this is more subtle and will depend on the specific tiling but we understand this case very well too.
The Tiling Space of $\mathcal{T}$
Now, let's also consider the dual topological theory to this setup. Whenever we have a group $G$ acting on a manifold $M$, we can consider the quotient space $M/G$ and so in the case of the translation invariant subgroup for a periodic tiling, we have the space $\mathbb{R}^2/\mathbb{Z}^2$ which is a $2$-torus $T=S^1\times S^1$ (in general, for $n$-dimensional periodic tilings we get the $n$-torus $(S^1)^n$). We'll call this space the tiling space of our tiling $\mathcal{T}$ and denote it by $\Omega_{\mathcal{T}}$ for an arbitrary tiling.
If we want to consider rotations as well, we need to be slightly more clever so that we don't lose too much information and instead consider the orbifold (stacky) quotient of the symmetry group. I won't spend any more time on the rotational version as it can get messy fairly quickly and would require its own new question in my opinion.
Now, how can we view the torus $T$ as an object which gives us information about our tiling? Well it's quite simple really; a point in the torus corresponds to a placement of the origin of $\mathbb{R}^2$ somewhere into our tiling. That is, if we choose some periodic tiling $\mathcal{T}$, including a prescribed origin, then if we move along one of the group elements in $G$ we should end up back to where we started and so locally, no matter how far away from the origin you look, everything about the tiling looks the same. We've effectively placed a metric on the set of all tilings of the plane by those tiles in $\mathcal{T}$ which says "if a translation by a vector $x$ can get me from the tiling $\mathcal{T}$ to the tiling $\mathcal{T}'$ then we say $d(\mathcal{T},\mathcal{T}')=|x|$, unless of course there is a vector of shorter length. So really we should say $$d(\mathcal{T},\mathcal{T}')=\inf\{|x|:\mathcal{T}+x=\mathcal{T}\}$$ and in fact this gives a well defined metric on the set of all tilings of the plane which contain only those tiles appearing in $\mathcal{T}$ and moreover this space is, by construction, homeomorphic to the tiling space $\Omega_\mathcal{T}$. This can therefore be taken as our definition of the tiling space.
What if $\mathcal{T}$ is aperiodic?
Well, this is the question you originally asked and it is a good question - it's the reason we spent so long defining the tiling space in terms of effectively local information of the tiling - remember for a periodic tiling the origin only needs to know whereabouts in a tile it is in order to determine what point that tiling represents in the tiling space. We're now set up to handle aperiodic tilings with only a small modification of the above metric.
First, let's suppose that $\mathcal{T}$ is a tiling of the plane which is aperiodic, so it has no translations that fix it. We can say this in notation as $\forall x\in\mathbb{R}^2\setminus\{0\},\:\mathcal{T}+x\neq\mathcal{T}$. Let's consider the following set of tilings. We'll say that $\mathcal{T}'$ is locally isomorphic to $\mathcal{T}$ if for all $R>0$, the set of tiles within $R$ distance of the origin in $\mathcal{T}'$, which we will denote by $B_R(\mathcal{T}')$, can be found somewhere in the tiling $\mathcal{T}$. In notation $\mathcal{T}'$ is locally isomorphic to $\mathcal{T}$ if $\forall R>0, \:\exists x\in\mathbb{R}, \:B_R(\mathcal{T}')=B_R(\mathcal{T}+x)$. We'll call the set of all tilings locally isomorphic to $\mathcal{T}$ the tiling space of $\mathcal{T}$ and write it as $\Omega_\mathcal{T}$. Notice I haven't told you the topology of the space yet.
Now, from the motivating material before, we want to put some metric on $\Omega_{\mathcal{T}}$ which in some way satisfies our notion of two tilings being close if they look very similar to each other around the origin out to a large radius. I'll put the metric here but note that it's not pretty - there are equivalent ways to define this metric and you can even forgo defining the metric altogether and just define the topology but none of them are particularly pretty - instead, just try to keep the intuition that two tilings are close if, after a little wiggle, they overlap each other around the origin on some very large patch of tiles. For tilings $\mathcal{T}',\mathcal{T}''\in\Omega_{\mathcal{T}}$ we define the tiling metric $d\colon\Omega_\mathcal{T}\times\Omega_\mathcal{T}\to\mathbb{R}$ by
$$d(\mathcal{T}',\mathcal{T}'')=\inf\left(\{\sqrt{2}\}\cup\{\epsilon\mid\:\exists u,v\: B_{1/\epsilon}(\mathcal{T}'+u)=B_{1/\epsilon}(\mathcal{T}''+v)\mbox{ and }|u|,|v|<\epsilon\}\right)$$ where $B_R(\mathcal{T})$ is defined as before to be the set of tiles within a ball of radius $R$ about the origin. It's a nice exercise to show that $d$ as above really is a metric and a slightly easier exercise to then show that if $\mathcal{T}$ is periodic, $d$ is equivalent to the previous metric defined in the periodic case. In this sense, the above metric is at least a proper generalisation of the periodic case.
Most of the hard work is done now. We have our tiling space $\Omega_\mathcal{T}$ and it seems to represent our intuition of what it means for two tilings to be similar to a 'high degree of accuracy' - there are hundreds of papers by the way on trying to understand the topology of these spaces - they are extremely complicated spaces. For aperiodic tilings which are locally finite (so there are only finitely many patches of tiles of a given radius appearing in $\mathcal{T}$) they are connected but not path connected spaces. They are also compact.
They are a bit like a solenoid as shown below, except they are not homogeneous spaces. Given a regularity condition known as repetitivity which I won't write here (a Penrose tiling is both locally finite and reptitive) a tiling whose tiles are polygons meeting edge to edge has a tiling space which fibers over the $2$-torus $T$ with fiber homeomorphic to a Cantor set. You can think of moving around the torus as translating the tiling by small amounts, and then hopping a small discrete distance from one point in the Cantor fiber to another corresponds to making a different choice of placing a tile far away from the origin, but everything else closer in than that tile is the same.
What about the algebra?
I will leave this section fairly vague as this post is already getting very long. As the astute reader may have guessed, our usual bag of tricks for studying these spaces won't really work - we have an uncountable collection of disjoint path connected components, each component of which is weakly contractible, so singular (co)homology and homotopy groups are all pretty useless in this setting. What we need is an invariant which can somehow see the connected components as opposed to the path connected components - for reasons I won't go in to, it turns out that the right tool is Čech cohomology.
For CW complexes the Čech cohomology is isomorphic to the singular cohomology so as a bonus, for periodic tilings, we can recover our original group $\mathbb{Z}^2$ as the first Čech cohomology group! For aperiodic tilings the Čech cohomology is a pretty good invariant, it gives a fair amount of information about the tiling space, which can in turn be translated into information about the tiling. We're still trying to find a full understanding of what information is really embedded in the Čech cohomology. Probably the biggest bonus of Čech cohomology is that we can actually calculate it for a large class of spaces including the Penrose tiling using various sophisticated techniques. The Čech cohomology of the Penrose tiling space $\Omega_P$ is given by
$$\check{H}^0(\Omega_P) \cong \mathbb{Z}\\ \check{H}^1(\Omega_P) \cong \mathbb{Z}^5\\ \check{H}^2(\Omega_P) \cong \mathbb{Z}^8.$$
There are a few other invariants we can associate to tiling spaces including its $K$-theory, the tiling groupoid, the $\lim^1$ invariant, various notions of dynamical equivalence and shape equivalence. For the reader who wishes to read more of what has been outlined here, I recommend the book by Lorenzo Sadun$^{[1]}$ linked above, and the seminal article of Anderson and Putnam (1998).$^{[2]}$
$[1]$ Lorenzo Sadun (2008) Topology of Tiling Spaces. American Mathematical Society, ISBN-13: 978-0-08218-4727-5. ISBN-10: 0-8218-4727-9
$[2]$ Jared E. Anderson and Ian F. Putnam (1998). Topological invariants for substitution tilings and their associated $C^\ast$-algebras. Ergodic Theory and Dynamical Systems, 18, pp 509-537.
I would like to come up with a final list of "tilings"
An ambitious goal. Grünbaum and Shephard, the authors of the book Patterns and Tilings, at first intended to make it just the first chapter of a book on geometry, but ended up with a 700-pages book just on this subject.
it appears that the terms "tiling" and "tessellation" can be interchanged
Yes, Grünbaum and Shephard say in p. 16 that the word tesselation is used "synonymously or with similar meaning" to "tiling"; English Wikipedia agrees.
Is there a clear separation between these two groups somehow?
One could draw the line in a variety of ways and degrees of generality. In particular, one could require the tiling to be invariant under some discrete group of isometries. In addition, one could require the tiles to be compact. In addition, one could require them to be polygons.
The sequence of triangles (p 3 2) change from spherical (p = 3, 4, 5), to Euclidean (p = 6), to hyperbolic (p ≥ 7).
Where did the (p 3 2) and (p = 3, 4, 5) come from, what does that mean? Also, in the Vertex figure, why do they use p and q and do q.2p.2p, instead of like the example of 3.5.3.5? Finally for this, why does the Wythoff symbol use a number 2 in there?
Wythoff construction is always based on a tiling by copies of a triangle, the Schwartz triangle; it is characterized by a (sorted) tuple of three positive integer numbers. That section of the Wikipedia article is an example, as follows from its title, they consider only tilings by right triangles, hence one of the numbers is always 2.
They mix two degrees of generality, assuming two arbitrary parameters $p$ and $q$ in the top three rows of the table but setting $q=3$ in the rest of the table and in the introductory paragraph. There is no good reason for that. However, the choice $q=3$ is motivated: $(6~3~2)$ is one of the few Schwartz triangle possible in the Euclidean plane. So if we fix $q=3$, then to list tilings by Schwartz triangles, we should consider $p=3,4,5,6,7\ldots$. So Schwartz triangles $(3~3~2)$, $(4~3~2)$, $(5~3~2)$ tile the sphere, $(6~3~2)$ tiles the plane, $(7~3~2)$ etc. tile the hyperbolic plane. These tilings are shown in the leftmost column. Each produces many different tilings depending on the particular Wythoff construction.
It shows a Coxeter diagram, Wythoff symbol, Vertex figure, and then p4m, [4,4], (*442), where does that come from, what is all that? For example on that last like (k uniform tilings), it has stuff like $[3^34^2; 3^26^2; (3464)2; 3446]$. Sometimes they will list several Wythoff constructions, like here, what does that mean?
These are different ways to describe the tilings.
The vertex figure describes what tiles we encounter if we travel around a vertex (each of the inequivalent vertices, if there are many). In particular, $[3^34^2; 3^26^2; (3464)2; 3446]$ means the following:
Schläfli symbol is another way to write down a simple vertex figure. It is best suited for the generalization for regular (where all vertices and facets are the same) polytopes in higher dimensions.
How to comprehend it all? Well... First of all, focus on some particular domain. For example, (temporarily) ignore all hyperbolic stuff, ignore stellar polyhedra, ignore higher dimensions, ignore Coxeter's bracketed notation, ignore everything unrelated to some chosen thing you're studying. However it is important to understand that polyhedra and tilings of the sphere are the same thing in this context. Wythoff construction is a thing that can be understood in an isolated way, so it can be a good start. Symmetry groups are an important concept, so it would be useful to study the platonic bodies and the wallpaper groups. Coxeter diagrams and (very closely related) Dynkin diagrams are a very big deal in many other contexts, so it may be a good idea to look at them, too, although this way it's easy to quickly go down the rabbit hole and get frustrated.
It's all for now, I am leaving the opportunity to recommend comprehensible books about tilings to others. Perhaps The Symmetries of Things is a good choice.
Best Answer
The difference is that you can have tilings where there are two vertices with the same sequence of tiles adjacent to it, but which do not map to each other by a symmetry of the whole tiling. Matching the vertex pattern is a local property, whereas mapping by symmetry is a global property.
Take for example the two 2-uniform tilings with the vertex patterns $[3.4^2.6; 3.6.3.6]$:
Pattern 1
Pattern 2
You can make a new tiling by combining these two tilings, simply by alternating some number of rows of one with some number of rows of the other. In this way you can eliminate some of the tiling's symmetries that map one row to the other, and so make it k-uniform for some larger k. Nevertheless, it will still be 2-archimedian since there are still only two vertex patterns.
For example:
The squares with one adjacent triangle are coloured red, and the others (zero or two adjacent triangles) are coloured blue. The red and blue type of squares alternates to infinity, so after a red row of squares the next row of squares will be blue, and vice versa. This makes it a periodic tiling.
Clearly no symmetry of the tiling can ever map a red row onto a blue row since the adjacent triangles can never match. Nevertheless, all the vertices of the red squares have vertex pattern $3.4.4.6$, as do all the vertices of the blue squares.