I can kind of grasp why this is the case as if we take the union of all finite subsets of cardinality $i$ as $i$ runs through every natural number, we are listing finitely many elements each time.
Whereas with the set of all subsets we can just keep adding new elements via Cantor's argument.
However I do not feel very secure on this type of understanding, I was just wondering if someone could deliver a good explanation as to why they aren't equivalent.
The first set is a strict subset of the second: if for example you consider the set of natural numbers $\mathbb{N}$, then the set of even numbers or the set of all natural numbers above a certain value, are all infinite sets and thus are only contained in the set of all subsets, but not in the set of finite subsets.
Actually the cardinality of the first is equal to $\mathbb{N}$, whereas the cardinality of the second (which is the power set or $\mathbb{N}$) is equal to the one of the real numbers.