I had the exact same symptoms as you, and I have the exact same environment (firmware 6.2, OSX Tiger, firewall disabled).
Software update didn't tell me I had any new firmware for my Airport Express, so I searched for how to update to 6.3. My AirPort Admin Utility was at version 4.2, but it looks like there's a new took available called "AirPort Utility", which is at version 5.3.2:
http://www.apple.com/support/downloads/airportutility532tiger.html
I installed this, and right away it found the 6.3 update for my AirPort Express. This totally solves the problem, and AirTunes now works correctly.
Hope this helps!