I'd succinctly say as others have posted here previously prime is a boxer's prime years physically and peak as a peak performance which they can't repeat or surpass.
Factoring in physical years doesn't overly work. Ron Lyle certainly didn't peak as a fighter prior to ever having fought professionally. We also have guys like Hopkins and Moore. The peak of a fighter is a reasonably compact period where he is at his absolute greatest in the ring. Some might keep up top form a bit longer than others. Prime is where is a usually much longer period where he is performing at a level pretty close to his best. Larry Holmes peak was in and around the Ali fight. His prime as a rough example may have been from, or just prior to Shavers 1 until perhaps Witherspoon He was losing a little but still did a lot of good thigns in that fight whilst not much after his decline was obvious. Hagler for me peaked around 82-83 and i reckon the Hearns war knocked him off his prime period.
Physical prime and peak are certainly not the same thing. Some obvious examples: Jersey Joe Walcott and Saoul Mamby, who were most successful after their physical prime.