What's the difference between Complex infinity and undefined?

Can somebody please expand upon the specific meaning of these two similar mathematical ideas and provide usage examples of each one? Thank you!

I don't think they are similar.

"Undefined" is something that one predicates of expressions. It means they don't refer to any mathematical object.

"Complex infinity", on the other hand, is itself a mathematical object. It's a point in the space $\mathbb C\cup\{\infty\}$, and there is such a thing as an open neighborhood of that point, as with any other point. One can say of a rational function, for example $(2x-3)/(x+5)$, that its value at $\infty$ is $2$ and its value at $-5$ is $\infty$.

To say that $f(z)$ approaches $\infty$ as $z$ approaches $a$, means that for any $R>0$, there exists $\delta>0$ such that $|f(z)|>R$ whenever $0<|z-a|<\delta$. That's one of the major ways in which $\infty$ comes up. An expression like $\lim\limits_{z\to\infty}\cos z$ is undefined; the limit doesn't exist.