Well, aside from the wood used, the finish material (lacquer vs. thick poly), and the quality of the neck-to-body joint, the first things you'd need to change are the bridge, the pickups, and the tuning machines.
Little stuff is of higher quality on the American Strat as well...nicer pickguard, better quality pots, better quality 5-way switch.
By the time you'd done all that, you'd have spent almost enough to buy an
American Strat, and it still wouldn't be the same quality.
Having said that, Squires CAN be made to sound a lot better with a few changes. After giving it a good setup, the pickups and tuning machines are the biggest offenders...changing them will improve the tone and playability substantially. Ebay can be a good source for American Series pickups, because lots fo folks like to experiment with aftermarket stuff. Also check the nut to be sure that the strings are well-spaced and cross the slot cleanly on the edge, that the saddles are adjusted correctly on the bridge, and that the spring tension is correct (tuning problems cause by incorrectly adjusted tremolo's are VERY common on Squire Strats).
If your tone controls aren't working, you have a wiring problem...go to
http://www.mrgearhead.com/ for wiring diagrams. If you're interested in doing your own setups and making any guitar sound better, buy the book
"Guitar Players Repair Guide" by Dan Erlewine.
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