I used Squier CV basses for years and was proud of the name on the headstock. Sometimes I would play at festival gigs and the soundman/other bassist would say 'wow' when they saw I was playing a Squier and not a Fender. My basses were modded to suit however with CS pickups etc, and I don't think there is really any discernible difference in tone. Essentially it's just a string vibrating within a magnetic field wired to a tone control and output jack.
However, having bought a few US Fenders recently I can say that they are absolutely better instruments all round. The necks on them are superb - I love the quartersawn maple, satin finish, rolled fret edges and lightweight tuners. The hardware on a whole is much better - the bridges have channels to ground the saddles for extra stability, the pickups are great and the basses themselves feel very lightweight.
Despite my stubbornness for years, I am now a convert to US Fenders