I think it was more than simple remorse. It was the realization that it had gone as Jesus called it. Why that leads to weeping depends on how you read it. If you read it as Jesus making the prediction based on his knowledge of Peter's character, then Peter is realizing that he really is as flawed as Jesus thought. If you read it as Jesus foretelling coming events, them perhaps the realization that he had been trapped by fate into this moment made him weep.