Are they considered different keys?
In terms of key signature, No. Since they use the same notes, the key signature will remain the same.
In terms of harmonic analysis, Yes! In terms of how we analyze the chords, it's much easier to read and make sense of when we analyze in the tonality of the piece as opposed to just the key signature, especially when using Roman Numerals for chord analysis. It's much easier to analyze a piece in A Minor than as relative to C Major.
For example: In A Minor, it's much easier to read an E Major chord as V, as opposed to V of vi (V/vi) that you would write it as when analyzing from C Major.
Is it a Key Change?
That depends! If the tonality switches to A Minor for only a couple bars, perhaps as part of a cadence, you might not necessarily change the key of your whole harmonic analysis. It may be easier to analyze that E Major chord as V/vi if it only happens for one or two measures. In this case, No, Probably Not.
On the other hand, if an entire section starts and ends in A Minor, the whole section is probably in A Minor, and would most likely be analyzed as such. Most modern composers and arrangers signify an apparent key change like this using a double bar line to show that this section is different and to pay careful attention. In this case, Yes, Almost Certainly!
This is definitely a question with a lot of nuance, and it really depends on your perspective and how you choose to analyze a piece.