When doing a "restore from backup" of my iPad 1 in iTunes, the restore process does not erase the new apps that were purchased after I created the backup. Also, the apps were purchased with a different iTunes account than the one I have authorized on my PC. Normally, I would get a message that tells me that those items are not able to transfered, and the restore process would delete them, along with any other apps/etc. that weren't part of the backup I created. Any ideas on why the new app survived the restore process? More importantly, how I can get items that are not part of the backup I created to erase?

The backup file does not contain apps, movies, books, synced photos, music, or just about anything that you can actually see in iTunes.

Here is the rundown on what backups do store, more: iPad: About backups

To remove apps via iTunes, go to the Apps link when the iPad is connected and uncheck all apps you do not want to sync, or have removed; then sync. Apps that you manually delete from the iPad should uncheck themselves the next time you sync.

The same holds for music, movies, or other media you want to remove. You change the sync settings, or delete them directly from the iPad.

I do not recommend deleting stuff from iTunes itself, unless you are really sure you don't' want it. Until iCloud comes along this is your only backup of some of your Apple music and movie purchases.