A lot of you don't get it, on the Heat, Ray's role is that of Robert Horry. And in game 6, that's exactly what he delivered, the clutch 3 to send it into overtime. That's Horry's career long signature play, team after team.
In Boston, however, he's suppose to be a part of some 'Big 3' consortium, which he clearly wasn't especially last season. Thus, him leaving makes sense. When one's past his prime, there's no point in hanging around, just to try to fit into some role which one isn't suited for. There was little left for Ray to do in Boston. He's a limited 3 pt specialist, who can play, at most some 15-20 minutes per contest.