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If you search the Corpus of Contemporary American English for these two expressions, you will see that minute by minute can be used as an adverb:
...as well as an adjective:
In both cases, minute by minute describes the pacing of some kind of progress (usually a flow of information).
Minute after minute is substantially less frequent, and its usage is slightly different. It can also be used as an adverb:
But it also has two uses where it functions as a noun:
In the construction "minute after minute of _"
In the construction "for minute after minute"
As you can probably gather from these examples, the emphasis here is not on the pace but rather the duration: minute after minute emphasizes how something continues without stopping.