What is the significance of limit points? When I had my first taste of topology a couple of years ago, our lecturer emphasized the following notions.


*

*closed set, closure, closure point

*open set, interior, interior point


Of course, these are all basically different ways of talking about the same thing. For example, from the family of closed sets of a space we can obtain its closure operator, and the fixed points of the closure operator are precisely the closed sets; the open sets are precisely the complements of the closed sets, etc.
Anyway, the concepts were well-motivated from the viewpoint of continuous functions between metric spaces. For example, it was demonstrated that if $X$ and $Y$ are metric spaces and $f : X \rightarrow Y$ is a function, then the following are equivalent


*

*$f$ is continuous in the sense of $\epsilon$-$\delta$.

*The preimage of a closed set under $f$ is closed.

*The preimage of an open set under $f$ is open.

*$f(\mathrm{cl} \,A)\subseteq\mathrm{cl}(f(A))$ for all $A \subseteq X$.

*$\mathrm{cl}(f^{-1}(B)) \subseteq f^{-1}(\mathrm{cl} \,B)$ for all $B \subseteq X$.

*$f^{-1}( \mathrm{int} (B)) \subseteq \mathrm{int} ( f^{-1} ( B ) )$ for all $B \subseteq Y$.


etc.
Anyway, on Wikipedia there's some related concept that weren't really dealt with in the course I took, namely limit points and isolated points. I understand the definitions (Wikipedia is clear enough), yet at the same time I don't understand their significance. They just seem like closure points, but less well-behaved.
For example, the set of all closure points of a set always includes the original set. The same is true of the set of all limit points, so long as the original set did not possess any isolated points, indeed we get the same answer. The only time we get a different answer is when the original set has at least one isolated point; but, in this case, the act of taking the set of all limit points is no longer so well-behaved; indeed, it will never include the original set.
So, its not clear to me the benefit of thinking in terms of limit points, as opposed to closure points. In what circumstances are limit points the right concept, and, more broadly, what is their significance?
 A: Well, like you, I wouldn't assign limit points quite the same conceptual status as say the closure operator or interior operator. But, it is indeed possible to define topologies using "limit point" as a primitive axiomatic notion. 
The key word is "derived set", which is actually an important notion. Given a subset $S$ of a topological space, the derived set $S'$ is the set of limit points of $S$. The key properties can be axiomatized: a (Cantor-Bendixson) derivative on a set $X$ is a monotone operator $\delta: P(X) \to P(X)$ that preserves finite unions (including the empty union $\emptyset$) such that $\delta^2(S) \subseteq \delta(S)$ and $a \in \delta(S)$ implies $a \in \delta(S \backslash \{a\})$, for every $S \in P(X)$. 
Then, given a derivative on a set $X$, define $S \in P(X)$ to be $\delta$-closed if $\delta(S) \subseteq S$. It may be shown this gives the closed sets of a topology, such that $\delta(S)$ is the derived set of $S$ with respect to this topology. In this way, there is a natural bijection between topologies and Cantor-Bendixson derivatives, so that the two notions are essentially equivalent. 
One reason not to fully accord derivatives the same fundamental status as (say) closure operators is that closure operators are far easier to work with in the setting of constructive mathematics (constructive topology), where for various reasons one might be interested in dropping the principle of excluded middle. 
That said, derived sets are very important. It was Cantor's study of iterated derived sets in the real line that led him to his theory of ordinals and cardinals. Not that I am expert, but they are used a great deal in descriptive set theory. 
Edit: Maybe the following will provide more of a "kick" for the OP. 
Problem: Show that every infinite closed subset of the real line has cardinality either $\aleph_0$ or $2^{\aleph_0}$. Thus, the continuum hypothesis holds at least for closed subsets of $\mathbb{R}$. 
I won't spell out the answer in detail here -- I'd really recommend trying to think for oneself how to prove it -- but suffice it to say that a critical notion in the standard solution is that of "perfect set": a subspace $S$ of $\mathbb{R}$ (or more generally, of any separable metric space) is perfect if every point of $S$ is a limit point of $S$. 
If you try this problem and get stuck, then try googling "Cantor-Bendixson theorem", and then see e.g. here for a nice theorem (theorem 1.1) that concerns the cardinality of perfect sets. 
The point is that it is limit point -- not closure point -- that is the "right" or appropriate concept for tackling this problem. Another problem along similar lines: show that a compact Hausdorff group is either finite or has cardinality at least $2^{\aleph_0}$. Again, it is the bifurcation of closure point into the separate notions of isolation point and limit point which is crucial to the analysis. 
