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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:

• cover these campaigns minute by minute
• planned out minute by minute

...as well as an adjective:

• a minute by minute narrative of the events
• a minute by minute account of his last critical hours

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:

• listened to this guy minute after minute
• stood there dazed minute after minute

But it also has two uses where it functions as a noun:

• In the construction "minute after minute of _"

  • recounting minute after minute of what he remembered
  • put minute after minute of science information on the screen

• In the construction "for minute after minute"

  • bounded ahead for minute after minute
  • fired and reloaded, often 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.