To start out, my favorites so far are Lords Of Thunder, Air Zonk (both for Turbografx 16), and the Thunderforce series, which is mostly on the Genesis/Mega Drive. I've played alot of shmups now but none of them stood out to me nearly as much as those did. I didn't grow up with shmups either, I've only started becoming a fan of them maybe 3 years ago. Meaning I don't really care for R-Type or Gradius. I respect them, but..meh. Lords Of Thunder and Air Zonk both I can never put down. Lords Of Thunder had an amazing soundtrack, great graphics for the time, and the gameplay is VERY intense and challenging, though it is possible to beat it. Just takes practice. Also I love the fantasy theme and the huge variety in not only level design, but gameplay elements such as buying upgrades and switching characters/elements. Air Zonk I love just because the art style rocks and..well it also has a very good bit of variety. Awesome weapons too, and you get a little sidekick to shoot alongside you as well. I also want to give a nice honorable mention to Bangai-O. The only one in the series (in my opinion) worth playing is the original, but man is it fun, and very unique. If you like this genre at all, or interested at all, or just want to blow shit up, you owe it to yourself to try it.
So yeah as far as my reasons for liking shmups, it's pretty simple. First of all, to me, it's the closest thing to an arcade experience that you can have at home, especially if you've got yourself an arcade stick like I do. The other one is that I don't always have the most time on my hands, but I still want something fun to play, and shmups are obviously very pick up and play friendly. Third reason is they are damn addictive if you find some you like. Fourth reason is they're always so creative with the gameplay (usually) and definitely the art style.
Another thing I like about shmups are the bullet hell ones. You know the ones that are so difficult and over the top that you'll probably never beat them? Sometimes I like a game that can kick my ass thoroughly just because it's fun to see how long you can last.
Screenshots below.
Thunderforce 6:

Psyvariar 2:

Bangai-O:

Lords Of Thunder (screenshot doesn't do it justice)

Air Zonk

For the record, you can find Air Zonk and I think Lords Of Thunder both on the Wii Virtual Console.