Actually, there is an active theory of algebraic topology for "pathological" spaces that has come a long way in the past two decades: Wild (algebraic/geometric) Topology. In fact, this has become a small field in its own right with a lot of recent momentum.
Descriptions of fundamental groups do become more complicated because in a wild space there may be shrinking sequences of non-trivial loops, which allow you to form various kinds of infinite products in $\pi_1$. Hence wild algebraic topology requires more than just the usual tools from algebraic topology but also is deeply connected to linear order theory, continuum theory, descriptive set theory, and topological algebra.
Here is an example of an astonishing result from this field, which addresses your interest in detecting homotopy type:
Homotopy Classification of 1-Dimensional Peano Continua (K. Eda): Two 1-dimensional Peano continua (e.g. Hawaiian earring, Sierpinski carpet/triangle, Menger cuber) are homotopy equivalent if and only if their fundamental groups are isomorphic.
The combined work of Greg Conner and Curtis Kent announced last year proves the same thing is true for planar Peano continua.
Once you realize how complicated these groups are due to the kinds of infinite products that can occur (although a word calculus of sorts does exist), it is absolutely remarkable that such theorems are true...almost scandalous. Results like the one above are very hard to prove. Eda's result required a lot of ingenuity and machinery that is being used and extended in current work.
Here is a little more pre-2000 history:
1950s - 1960s: There were a few scattered papers by some prominent mathematicians, e.g. Barrat/Milnor, H.B. Griffiths, Curtis/Fort.
1970s: Shape theory was developed to extend homotopy theoretic methods to provide invariants for more general spaces. The idea of space theory is to understand objects as (or at least approximated by) inverse limits of the usual "nice" spaces, applying your invariant to the nice approximating spaces, and call the inverse system of algebraic objects a "pro-invariant" and the inverse limit a "shape-invariant." The book Shape Theory by Segal and Mardesic is, I think, the best book on this topic. However, shape invariants only sometimes help with understanding homotopy type and traditional algebraic invariants of wild spaces.
1980s: Not much happened except for Morgan and Morrison fixing H.B. Griffiths description of the fundamental group of the Hawaiian earring.
1990s: Katsuya Eda, whose background was in logic, discovered that the Fundamental group of the Hawaiian earring behaves like a non-abelian version of the famous Specker group $\prod_{\mathbb{N}}\mathbb{Z}$. Eda was the first to make the key connection to order theory and describe the Hawaiian earring group as a group of reduced linear words $w$ (like a free group) where $w$ has countably many letters and each letter of your alphabet can only appear finitely many times in $w$. This work made the Hawaiian earring group practical to use; it is the key to many recent advancements.
Since Eda's work there has been a great deal done and there is now a huge amount of literature on the subject.
Do a good reading of that Wikipedia page: path-connected (usually) means that for every $a\neq b \in X$ there is a path from $a$ to $b$ (a continuous function from $[0,1]$ into the space $X$ with $p(0)=a$ and $p(1)=b$ is called a path from $a$ to $b$), while for an arcwise connected space $X$ there is the stronger requirement that there is such a $p$ but that also is an embedding (so that $p:[0,1] \to p[[0,1]]$ is a homeomorphism). So an arcwise connected space is always path-connected but the reverse sometimes does not hold (there are finite counterexamples, e.g.; obviously an arcwise connected space must be uncountable, while a finite space can be path-connected: a continuous map is a lot weaker than an embedding)
A theorem (not trivial though, it depends on the study of Peano continua) proves that if $X$ is Hausdorff (as is very common, e.g. all metric spaces, ordered spaces, many topological vector spaces etc.) then $X$ being path-connected implies $X$ is arcwise connected too. This can be quite useful to know.
So in practice these notions coincide, but there are marginal examples where the notions differ.
Maybe for your first author path actually means a homeomorphic copy of $[0,1]$ and he really defines what most texts would call arcwise connectedness. Check his definition of path.
Best Answer
As others have pointed out, every arcwise connected space is connected.
$\pi$-Base, an online version of the general reference chart from Steen and Seebach's Counterexamples in Topology, gives the following examples of spaces that are connected but not arcwise connected. You can learn more about these spaces by viewing the search result.
A Pseudo-Arc
An Altered Long Line
Cantor’s Leaky Tent
Closed Topologist’s Sine Curve
Countable Complement Extension Topology
Countable Complement Topology
Countable Excluded Point Topology
Countable Particular Point Topology
Divisor Topology
Double Pointed Countable Complement Topology
Finite Complement Topology on a Countable Space
Finite Excluded Point Topology
Finite Particular Point Topology
Gustin’s Sequence Space
Indiscrete Irrational Extension of the Reals
Indiscrete Rational Extension of the Reals
Irrational Slope Topology
Lexicographic Ordering on the Unit Square
Nested Angles
One Point Compactification fo the Rationals
Pointed Irrational Extension of the Reals
Pointed Rational Extension of the Reals
Prime Ideal Topology
Prime Integer Topology
Relatively Prime Integer Topology
Roy’s Lattice Space
Sierpinski Space
Smirnov’s Deleted Sequence Topoogy
The Extended Long Line
The Infinite Broom
The Infinite Cage
The Integer Broom
Topologist’s Sine Curve
Uncountable Excluded Point Topology
Uncountable Particular Point Topology