Software Update on my desktop was erroneously saying that all of my software was up-to-date yet iTunes was saying that it needed a later release. Our other computers had the current iTunes release.
We went into our administrator account on my desktop computer, downloaded the iTunes install package from Apple (so, yes, this was downloaded separately and not in Software Update - http://www.apple.com/support/itunes/install/ ), and installed it. Hopefully it will now be included (as it had been) in Software Update.
I'm running Mac OS X 10.7.4 now with iTunes 10.6.3.
My desktop Mac is old enough (Mac Pro 1.1 bought in Fall 2006) that I can't upgrade to Mountain Lion. The upgrade list at Apple ( http://www.apple.com/osx/how-to-upgrade/ ) lists dates rather than model numbers. The date info is not readily available from About This Mac or System Information.