I don't think it does, but I have a Jacoby break cue that smashes the balls better than any other cue I have tried and it has a big, healthy butt. It is much bigger in diameter than my playing cue.
Many a game is won on the break, breaking with a poorly hitting butt doesn't make sense.
Hu
The three key ingedients in a well built cue are tip, shaft, and butt.
Hu...You already KNOW this. Technique is much more important than the construction of the butt (it's a piece of wood afterall). Perhaps at the elite level one pro can tell a slight difference between butts, on a break cue...but the "feel" of something is so subjective from person to person, that the OP's question, I thought, was directly at the general group of players...so NO, it doesn't matter. I have seen champions break with a house cue, with great technique, timing and results. I have also seen players break poorly, because of poor technique and timing, no matter what cue they break with.
Scott Lee
www.poolknowledge.com