Generally, flats don't sound as bright as roundwounds. They don't have that zing that new roundwounds do, but you also get much less finger-squeak. Some flats are brighter than others - D'Addario Chromes have a reputation for being pretty bright when new.
A lot of the modern bass sounds - Stanley Clarke, Marcus Miller, Chris Squire, Geddy Lee, John Entwistle (the originator of rounds for bass!) - are very bright and require rounds, and I think that's what many of us are used to. Flats are a different sound altogether. Actually, Steve Harris of Iron Maiden uses flats, which always surprised me, because his sound has a lot of treble click but he also changes them every show! Phil Lesh used flats in the early days of the Grateful Dead, which also surprised me, but it makes sense considering he was co-inhabiting a musical space that included two guitars and keyboards. I think the classic flatwound sound would be Duck Dunn, James Jamerson, etc.
Just some rambling, hope it helps!