What is the significance of limit points?

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?