In my experience, playing bass makes you more aware of how all the instruments fit together, and what the song needs. Ever notice how many bassists are producers? Or how many bandleaders are bassists? Bassists listen to EVERYONE, not just themselves (ducks the tomatoes thrown by guitarists). I'm not saying ALL bassists do that, or that NO ONE who plays guitar (or whatever) pays attention to the other instruments. But in general, I think the bass plays a supporting role in a band, and that tends to make bassists more aware of what's going on around them.
In most of the bands I've played in, I have been the "arranger" (of sorts), and the guy that everyone looks to for dynamic cues, endings, and whatnot. Over the years, I have really become aware of the importance of playing FOR THE SONG, rather than playing what makes me sound cool. When I was younger, I wanted to play all over everything. As I've gotten older, and gotten more experience playing different styles, I have learned that sometimes the part that is best for the song is the simpler part.
A somewhat astonishing example of that philosophy is the song "All Right Now", by the band Free. Paul Rodgers (later in Bad Company) sang it, and I believe wrote the lyrics. The music was written by Andy Fraser, the bassist, who was only 17 or so at the time. Now, if I had written a song when I was 17, there would be bass ALL OVER THAT SUCKER! Andy Fraser, however, showed remarkable restraint. Other than the really cool bass riff during the guitar lead (that bass riff is pretty much a solo in itself), the ONLY bass guitar in the whole song is during the choruses. There is NO bass during the verses. NONE. So when it does come in, it sounds HUGE! Of course, the fact that Andy Fraser was a classically trained pianist from a young age might have helped....
Anyway... yeah. Being a good bassist means being a good listener!