beautiful proofs, or perhaps more 'elegant' than beautiful, can be short and direct, and don't involve unnecessarily difficult ideas. sometimes people "confuse" (there is no arbiter, I'm just trying to offer *some* distiction) novel with elegant.
I myself can't decide if the proof that there are infinitely many primes using a topology on Z defined by arithmetic sequences is genuinely elegant or just novel, or if it can't be both. But then the proof of hte fundamental theorem of arithmetic (every polynomial of C has a root in C) using homotopy theory has to be elegantly beautiful doesn't it? Or is it a matter of sophistication?
One thing is for sure, any induction proof has to be inelegant and ugly: it offers nothing insightful to the proof or the result - knowing it true for a trivial case, and inducting on a presumption doesn't seem to offer any illumination.
Perhaps we might take the view that if the argument is one smooth flowing thread that that the proof is elegant, but then that might just come down to the presentation of the author. And are we distinguishing between an elegant result (exp(ipi)+1=0) or an elegant proof of a result?
Here is a case which is an elegant result and (to my mind) is often presented with an ugly proof:
Sylow's Theorems. If G is a finite group of order mq, with q a prime power and m prime to p, then there is a group of order q in G, moreover, all subgroups of order q (the sylow subgroups) are conjugate, every p subgroup is contained in some sylow subgroup the number of sylow subgroups is congruent to 1 mod p (I think, i can never remember that bit correctly).
that result gives you a tremendous amount of information about a group, tells you so much about why groups are so powerful, and is only a couple of lines long, yet its usual proof is horrendously dull. i can think of an elegant proof (of soem parts) using vertices and sources, but you need to know a lot more about groups and their representations before that becomes applicable.
undoubtedly the 4 colout theorem and the classification of finite simple groups are ugly proofs as they are just checking proofs, they offer no ingenuity. ok, that's overly dismissive, since 4-colour required ingenuity to show that all cases could be reduced to one of the computer checked cases, and I'm sure that in CFSG there are bits of ingenuity for different cases, but over all it's a checking proof. But, such are the clouds obscuring all these objects, right now that's the best we can do.