The different sizes of 'infinities' mostly deals with sets and series. But consider this: A set (or series) contains bounded information. Example, X={all integers}={1,2,3,4,5,...}. Y={all prime integers}={1,2,3,5,7,11,13,...}. Both sets go inot infinity. But if you took X, and removed Y from it, there would be less in X (but still going to infinity). If all infinities were equal, then setX minus setY would equal 0. But that is not really the case. And since what results from taking Y out of X leaves a set that is diminshed from what set X was, then how can that infinity be the same? This is the problem with working with sets and series.
To clarify what the meaning of infinity is, just think of something that goes on forever. Infinity means mostly to 'keep going'. There really is nothing equal to infinity. Most often, mathemtaics uses the term "as it goes to infinity". Not "when it equals infinity". Sets and series by concept mean something that is bounded. Even if the series has an infinite amount of possibilties, it is conceptually bounded by something :example: the set of integers.
Infinity is a concept. It is not a mathematical rule.