Both have similar strengths in being incredibly technically sound and being able to carry lesser talent to great matches. But I think Bryan trumps Bret in near every category.
Peaks and volume Bryan runs away with it. I know that Bryan has the advantage of working with a ton of great workers, but he also rounds out his resume with legitimately great matches against the likes of Miz, Wyatt, and Triple H. And while Bret has some great matches, Bryan's peaks are higher.
When it comes to versatility, Bryan might be better than just about anyone (I'd consider arguments for Funk). Fiery underdog, ruthless heel, pissed off ass-kicker, and even cowardly heel at times. Almost every great Bret match I've seen has him as a babyface or defending champion.
Both are terrific offensive wrestlers, but Bryan's ability to wear more hats I think offers more variety than Bret. Selling is really the only area I don't have a strong argument in favor of either.