Showing two topological spaces are not homeomorphic 
Having infinitely many open sets in a topology is a topological property

#1: What are example of two topological spaces, preferably simple ones, which this statement show are not homeomorphic.
But, more importantly why does it show this? Having trouble understanding topological properties in connection to homeomorphisms.
#2: What about two spaces where this statement doesn't help deciding whether they are homeomorphic or not?

If $X$ and $Y$ are topological spaces with respective topologies $\tau_X$ and $\tau_Y$, and if $f : X \to Y$ is a homeomorphism, then $f$ induces a bijection $\tau_X \mapsto \tau_Y$: each $U \in \tau_X$ is mapped to its image $f(U) \in \tau_Y$ defined as usual by the formula 
$$f(U)=\{f(x) \,|\, x \in U\}
$$ 
The proof that this map $\tau_X \to \tau_Y$ is a bijection is simple, and uses only the definition of homeomorphism.
As a consequence, if $X$ and $Y$ are homeomorphic then the two sets $\tau_X$ and $\tau_Y$ have equal cardinalities. So, for example, it cannot happen that one is finite and the other is infinite. It also cannot happen that one is countably infinite and the other is uncountably infinite.
