It makes a difference in the high end of the tone....tends to cut thru better than with dead strings. That percussive slap of the strings on the frets comes thru and gives better articulation of the pick (or fingers) hitting the string. Fieldy from Korn is an extreme example of this element - although I don't particularly like his tone. I love Trujillo's on "Art of Rebellion".
I always throw on a new set for a proper session but when they start dying I replace them and keep the old set, then boil them later for pre-production demos. Saves a bit on the string bill. When I was gigging a lot and couldn't afford strings they used to get me through a gig or two before having to be replaced. The more you boil them (or deaden them) the less effect it has though.
Having a decent bass usually means you can get away with leaving them on a bit longer. Most music stores have new high-end strings on all the basses including the cheaper ones to make them sound better. Give it an hour of solid sweaty playing and you'll find out exactly how bad (or good) it really sounds.