Variables : A variable is a quantity that may change within the context of a mathematical problem or experiment. Typically, we use a single letter to represent a variable. The letters $~x,~ y~$ and $~z~$ are common generic symbols used for variables. Sometimes, we will choose a letter that reminds us of the quantity it represents, such as $~t~$ for time, $~v~$ for voltage etc.
Parameters : A parameter is a quantity that influences the output or behavior of a mathematical object but is viewed as being held constant.
Arguments : The word argument is used in several differing contexts in mathematics. The most common usage refers to the argument of a function, but is also commonly used to refer to the complex argument or elliptic argument.
An argument of a function $~f(x_1,...,x_n)~$ is one of the $~n~$ parameters on which the function's value depends. For example, the $~\sin x~$ is a one-argument function, the binomial coefficient $~\binom{n}{m}~$ is a two-argument function, and the hypergeometric function $~_2F_1(a,b;c;z)~$ is a four-argument function.
Note: In general, mathematical functions may have a number of arguments. Arguments that are typically varied when plotting, performing mathematical operations, etc., are termed variables, while those that are not explicitly varied in situations of interest are termed parameters. In some contexts, one can imagine performing multiple experiments, where the variables are changing through each experiment, but the parameters are held fixed during each experiment and only change between experiments. One place parameters appear is within functions.
Examples :
Ex -$\bf(1)~:~$ A function might a generic quadratic function as $$~f(x)=ax^2+bx+c~.$$
Here, the variable $~x~$ is regarded as the input to the function. The symbols $~a,~ b ~$and $~c~$ are parameters that determine the behavior of the function $~f~$. For each value of the parameters, we get a different function.
Ex -$\bf(2)~:~$In the standard equation of an ellipse
$$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1~,$$
$x~$ and $~y~$ are generally considered variables and $~a~$ and $~b~$ are considered parameters.
The decision on which arguments to consider variables and which to consider parameters may be historical or may be based on the application under consideration. However, the nature of a mathematical function may change depending on which choice is made.
Equality, in many system and particularly for first-order logic with equality, is a relation of the logic. Most logical systems don't have a mechanism for making a definition. It simply makes no sense to say $(1:=S(0))\land P$ in these systems. A definition isn't a claim. It's not a statement that has a truth value.
So, not only are equality and definition not the same, they aren't even the same sort of thing. The most common way to handle definitions is meta-logically and informally. A definition like $1:=S(0)$ is an instruction to you, as the reader, to mentally replace $1$ with $S(0)$ everywhere you see it. You then trivially get $1=S(0)$ be reflexivity, because this really means $S(0)=S(0)$.
There are several ways of formalizing the notion of definitions. The typical way would be an extension by definition. For this perspective, a (meta-logical) definition like $1:=S(0)$ means: "Add the constant $1$ and the axiom $1=S(0)$ to the current theory, then continue with this new theory." Note how operational this is. It's instruction to create a new theory. "Add the constant $1$ and the axiom $1=S(0)$ to the current theory, then continue with this new theory," is not a statement that is true or false.
Some logical systems have an internal mechanism for making definitions. One example is the system used by Coq. The $\delta$-reductions do a formal version of the "mental replacement" described earlier. To over-simplify, it basically states that when you are in a context with a definition like $1:=S(0)$, then $1$ reduces to $S(0)$. The typing judgment is (supposed to be) sensitive only to normal forms of the reduction relation. There are constraints on the rules that introduce definitions to avoid (or clearly define the behavior of) multiple, potentially inconsistent definitions.
Best Answer
Generally, I wouldn't say there's a difference, other than one term being favored over the other, in certain fields.
Mathematically, a sequence/list of elements of a set $A$ is just a function from some subset of the natural numbers, to the set $A$. Symbolically, a sequence/list is just a function $f: D \to A$, where $D \subseteq \mathbb{N}$.
Generally sequences are infinitely long; they map all of the natural numbers to elements of $A$. Most lists I encounter are finite, but there's no reason they need to be. The term sequence is almost always referring to the kind of sequence that analysts use (where the items in our sequence are points/subsets of some topological space), while everyone else has to refer to an ordered collection as a list, so people don't think we're talking about analysis.