It depends on who you ask. Some musicians swear up and down that one tonewood is far above the rest. While the woods and materials involved do affect the sound, its so slight that most people could never tell the difference.
The pickups and strings are what really control your sound on the electrics, as does other factors such as pickup height, string gauge/type, etc...
As far as playability, thats purely on the build of the guitar, as a poorly built guitar will feel and play like a poorly built guitar. It may sound just as good as that $4000 custom built top-of-the-line premium guitar, but it will NEVER feel like it.
The only thing that the 'Body' consists of, is a shaped slab of wood with no hardware in it. Hilmar was correct to list each part separately because they're all unique parts of the guitar(making a special case for neck-thru guitars, where the body includes the entire neck), and as such, fulfill unique tasks. Your question is somewhat broad, but not unreasonably so.
You won't notice so much difference in tone between woods, only your sustain (Although, string age and bridge construction both factor in as well) will really be affected, but its hard to pinpoint it as just the tonewood or just the bridge, or just the strings, as they all weigh in on the equation.
Basically, with the right parts and enough patience, you can make a piece of plywood sound close to a $4000 guitar, but your hands will absolutely notice the difference, where a listener won't hear it, especially not one who hasn't trained their ear.