A gold-eared bassist would be hard pressed to tell the difference between woods, but solo, I think I could probably tell the difference between a Squire and a $2,000 American Fender. trying to cop the same general tone/vibe. In a band, the difference is obviously less noticeable, but I've switched between my main bass (MOJ ESP P/J clone) and a backup bass (Squire VM P bass) and noticed that the nicer ESP "fits" in the mix better.
But about 99% of the difference comes down to a few harder to quantify issues. First, I'd say consistency of sound is much better in more expensive basses; cheaper basses might have dead spots, or loud spots, or weird frequency bumps. Second, more expensive basses stay in tune longer and their necks stay true longer; tone/volume pots stay less scratchy, there is less hum, etc, so from gig to gig, song to song, note to note, every sounds smooth and even and "high quality." Lastly, a more expensive bass tends to just "feel" better to me, which makes my playing easier; I'm worried about the dynamics in the bridge and nailing that fill, not fighting the bass's weird neck shape or rough frets. So yes, I might "sound" better, but that's only because I'm not a good enough musician to fight the bass and sound good at the same time!
That said, I have 0 problems gigging and playing with a $200-300 bass, though under that amount I'd probably have some issues. The difference between a $1,000 and a $2,000 bass, though, is likely minimal; other than the "cool!" factor of the look/wood, or very nice 3-band active electronics, I'd say there is nearly zero sonic difference between the two.