The most prestigious competitions are the larger ones with some national exposure. The AHA National Homebrew Competition is the largest and most prestigious, by far. An NHC medal means a lot. I do not know where the Longshot competition fits in, really, as it is still rather new.
The larger single-site competitions come after that. I do not know them all, but the Dixie Cup, Bluebonnet Brew-Off, and Upper Mississippi Mash-Out are among the largest and tend to rotate through the #1 spot. The larger competitions also provide the largest variety of categories. Just for full disclosure, my local club sponsors the Dixie Cup, and I organized it last year.
As for feedback, that's a bit of a crapshoot, no matter where you enter. Sometimes you get judges that are good at feedback, sometimes not. Your best bet for feedback probably is in the smaller competitions in areas where BJCP judges are concentrated. That said, I have seen many, many good scoresheets come from the Dixie Cup and Bluebonnet.
As for pride, well, I would certainly be very proud of a medal from NHC. I also would treasure a Dixie Cup medal, since that's my club. Shoot, I would be proud of just about anything, though.