There are some sounds that you simply can't get with rounds. As I say above for example, the sound of flats with a pick with the highs up so the attack is pronounced. Very different with flats than with rounds. With flats you get the click/thunk on the attack, and then the tone is "softer" or "rounder" after the attack. It's one of the classic bass sounds, actually--think Carol Kaye, gobs of 60's/70's stuff, "cop show" bass, etc.
With rounds with a pick and the treble/tone dialed up, well you obviously get the zing or "buzz" in the note, even with old strings. Live in a setting where it's not that demanding you can cop the flatwound vibe for sure (I do it all the time with dead rounds and a foam mute) but it certainly sounds different. In the studio there's no substitute for flats if you're going for that sound. To think that flats are simply all about a bassy tone with no high end is overlooking what really differentiates them from rounds.