The KPA and AxeFx amp's work very different.
Number of Amp models:
The KPA can use unlimited numbers of profiles of real amps (1000 can be stored in the KPA at the same time).
The AxeFx has a limited number of dedicated models for each included amp.
Accuracy:
The KPA's profiles are VERY close to the sound of the used real amp, cabinet, mic, mic-postion, preamp ... can be A/B'ed by the profiler
The AxeFx may be close to the used amp, cabinet, mic, mic-position, preamp .... can only be A/B'ed by the manufacturer
Different amp settings:
In most cases need the KPA a new profile for different amp settings
Even while the KPA has knobs for drive, and tone - it can sound very different when compared to the real amp - when you change any of these settings.
Profiles of different amp channels, drive setting, and amp switches (e.g. bright/normal) are needed to capture the full amp.
AxeFx model can be used for more amp settings e.g. different drive settings or tone stack settings.
Even the AxeFx has no full amp model. The AxeFx needs a new amp model for each amp channel or mode.
More amps:
KPA - just profile the amp or search for free or commercial profiles of that amp
AxeFx - wait for a firmware update
Tweak the amp's:
KPA - has all the basic options (drive, tone stack) and some advanced parameters
AxeFx - has MUCH more options