I've got a set of each - a 59 set on an alder body and a 63 set on a swamp ash body
They are both really good choices for the "classic" strat sound, with loads of dynamics, and that really good woody neck sound, which is what I was after. They are quite similar in most ways, on the guitars I've got them on the 59s have a slightly warmer sound and the 63s sound slightly more focussed, but some of that will be because one's an alder body, ones ash.
I would say the 59s sound really SRVish to my ears, the 63s seem to have a slightly more mid scooped feel, but it's a subtle difference. Either way, they are both great sets and you won't be disappointed with either of them.