Just some clarification of terms (which can get fuzzy!):
Jacket BCD - the aircell is the whole vest (or nearly so) and wraps around you. That puts some of the air in front of you when fully inflated. Helpful if you want to have your head and neck fully out of the water at the surface, but means the air cell will also hug or squeeze you as it inflates. It's also just a lot of "Stuff" on the front of your body (built in pockets and such).
BPW - Made up of components. The Backplate (which can be aluminum, stainless steel, plastic, or even a fabric "soft-plate"), the Harness (ranges from 1 piece continuous to fancy multi buckle padded monstrosities), and the Wing (donut or horse shoe air cell that goes between the plate and tank). Minimal clutter on the front of you, pockets and D-rings can be added and put where you want them, and the air cell is behind you (typically good for in water trim, not as good if you want to sit high in the water at the surface. With my BPW I typically hang out on the surface with my chin touching the water... if I want to sit higher then I lay back on the wing).
Back inflate - Basically a jacket where the air cell is strictly behind you. Think prebuilt pockets and such of a jacket with the air cell of a BPW.
As to benefits of each, some is addressed above.... but the most critical thing is to find what YOU prefer. I love the freedom of minimal clutter that a basic BPW gives me. My wife can't stand back inflate or BPW, only likes Jackets. Since your buddy gave you a jacket BCD you have a chance to dive it a bit and see what you like or dislike about it. Then see if you can borrow a BPW for a few dives. Then borrow or rent a back inflate. It really is all about personal preference until you get into niche stuff (like tech diving or vintage) where the dives dictate the gear.
Respectfully,
James