I understand the differential cryptanalysis up to the "finding the last subkey" part. If XORing with the key doesn't change the differentials, how can testing different key affect the equations we found?
EDIT: I think I kinda get the answer now. Although I'd like a confirmation:
After getting all those probabilities on the differentials at different stage in the cipher. You test these probabilities with real pairs of messages and ciphertexts and you see if for an hypothetical key the probabilities you found hold. If they don't it's probably because you had a bad key, if they do then you probably have a good key.
EDIT2: Mmm, actually no I still don't understand. Even with a bad key, probabilities we found still should apply...