What difference can you expect when playing with flat-wound vs round-wound strings on an electric bass/guitar?
I know that they feel different - I am asking explicitly in terms of sound.
Roundwound strings are brighter, due to a higher presence of upper-order harmonics, while flatwounds are less bright. This occurs because roundwounds are more flexible than flatwounds.
Imagine a string threaded with tightly packed beads. If the beads are spheres, then you'll be able to bend the string relatively freely; if, however, the beads are cubes, then the edges and corners of the cubes will inhibit the bending of the string.
This same principle applies to roundwound vs. flatwound strings. When you pluck the string, it begins to vibrate, and as it vibrates, it flexes. Roundwound strings will have greater flex, while flatwounds will have less, because the flat windings inhibit the free vibration of the string. This suppresses the upper harmonics, causing flatwound strings to sound more mellow.