One interesting difference is that if you are learning a proof, then you know when you are successful. At least for my research, I generated a bunch of small c

One interesting difference is that if you are learning a proof, then you know when you are successful. At least for my research, I generated a bunch of small cases of the theorem I was trying to prove, then tried to learn the general case. If I was successful, I would have a proof of the original conjecture. When learning programs from some data, you can always test your program against the data but never can be sure that you are generalizing the data properly.