What are the advantages/disadvantages? I've heard having a steep angle will put more tension on the strings and will let them sit in their saddles better when doing bends.

I'm wondering because I want to install a bigsby on my 339. The guy I got it from threw in a b3 bigsby when I asked for one. Bigsby recommends the b7 or b12, but said I could install a b3 if I wanted. Just wondering how important the tension bar is, or if there's any advantage to NOT having one.
I can't think of any benefits off the top of my head of a shallower break angle at the bridge. Steeper is generally better, since it puts more pressure on the bridge and transfers more vibration. I would really strongly suggest using a B7 instead of a B3. The B3 is a compromise on what's already a touchy design, and it shows.

Putting a bigsby on your 339 is going to add some low end and will probably fatten up the sound a bit. I really like what bigsby units do for Gibson style guitars. I'd suggest getting a vibramate though, drilling one of those isn't something you should do if you can help it.