Update 十月 2017: As of iTunes 12.7, the entire concept of backed-up apps has been removed from iTunes, so you should no longer see any prompts about backing up apps when syncing or backing up in iTunes. Apple is supporting a version of iTunes through early iOS 12 that supports a Mac/PC iOS app library and installing apps through iTunes but this version of iTunes is not compatible with macOS Mojave, nor do we expect it to continue to be supported moving forward.
Here's the original post information, should you need it for iTunes 18.104.22.168 or earlier.
If you click the Back Up button in iTunes, you'll most likely see the following dialog:
What does this mean? And, like all dialog boxes iTunes shows me, should I click the no button like I'm inclined to? :)
Short answer: Yes, you should back up your apps in iTunes.
Long answer: As part of the normal iPhone backup, your data saved within each of the apps is backed up, but not the actual application itself. Backing up apps in iTunes means that iTunes is going to copy the application themselves (not your data, but the files that make the app run) to your iTunes library. This used to be critical before you could download past purchases from iCloud on your device, but it is still useful. If you later need to wipe and restore your iPhone from a backup, iTunes will install your apps back onto your restored phone, but only those apps backed up in iTunes. For the apps not backed-up in iTunes, you'll have to remember what apps you had and download them from the app store (and/or iCloud) again. (Edit: as of iOS 9, the apps that were installed at the time the backup was made are downloaded from the App Store and reinstalled after restoring the backup if you sign in with the same Apple ID during setup.)
If you choose not to back up your apps in iTunes, the restore process will still put your data back on the iPhone for those apps, it just won't install the app for you. Once you download the app and open it, your data will be there waiting for you.
If the idea of remembering what apps you had installed and manually installing each of them after a restore sounds tiring, then yes, you should choose to back up your apps in iTunes. But, if you're always running out of disk space on your Mac/PC, you may want to trade the possibility for future hassle for precious megabytes now.