Experience and experiment are different and even conflicting modes of knowledge. The clearest difference is communicable certitude – experiments are supposed to be set up with perfect clarity in advance with clear criteria that could be verified even by machines or other ignorant beings; but experience is a non-theoretical awareness that can be had even by brute animals and so needs no abstract, communicable element. Again, an experiment is a more or less decisive event that is meant to serve as a paradigm instance of the whole while experience has no one decisive event but is composed of a messy series of mere facts. Think the difference between the gardener and the botanist, the good wife and the successful marriage therapist, the engineer and the scientist.
The opposition between the two is exacerbated by our wish to transcend the opposition. It would be nice if the resolution to all problems were potentially theoretical and therefore communicable or “publicly verifiable” or even “empirically given”, but experience is both too subtle too stupid for that. It would also be nice, one supposes, if experience alone could be decisive, but it is itself motivated to a large extent by the desire to organize itself into theory. There is also an unavoidable finesse of experiment itself, which might be responsible for why so few of them are really repeatable.