Yes, this is possible. The backups themselves contain application data and other information that isn't particularly version-specific (with the exception of going from 4.3 backwards, as pointed out by hobs above).
In fact, I had to do this exact process recently, going from 4.2.1 to an iOS 5.0 gold master on an iPhone using a backup, and it went off without a hitch. Developers who want to install a prerelease OS version must put their devices in recovery mode, clean install the OS, then restore application data and settings from a backup. I did this for my iPhone, which I hadn't upgraded to the latest 4.x OS, and everything came across cleanly to iOS 5.0.
The one thing to watch out for is that if you don't encrypt your backups (by selecting the "Encrypt iPhone backup" option in iTunes), your device's keychain will not be transferred to the new device. For non-encrypted backups, the keychain is only preserved when restoring onto the same device you backed up from.
Ok, I got it done, but it took a little bit of a run around.
I had failed to mention in my question that the size of the original 16GB backup was indeed close to 16GB. Clearly not going to fit on the new 8GB phone.
The very first time I tried to restore from backup, I got a message about the size incompatibility. Every subsequent time it would just hang at "Estimating Time Remaining..."
Next, I decided to remove much of the superfluous data on the old phone, to get a new backup of it under 8 GB. Trying to use this new, now smaller, backup resulted in the same hang up.
At this point I thought it just might not work.
With nothing to lose, I decided to have iTunes restore the phone to factory. Then, the very first thing I did was try to restore from the new smaller backup... Success!!
Somehow the failed attempts before hand, were keeping the phone from successfully restoring from backup.
Best Answer
This used to be possible up to iTunes 12.6
Individual .ipa files for each app were stored on the user's local computer & could be restored to a new device [assuming they were capable of running on it].
Since 12.7, iTunes no longer keeps copies of .ipa files & after a new device install, apps are downloaded direct from the App Store - these will always be latest versions.
Apple did keep a special corporate version of iTunes available - currently 12.6.5 - via this link; Deploy apps in a business environment with iTunes but it requires you already have the .ipa files on your computer, or can pull them from the old device. I haven't tried this as a working method in a long time so can't quite recall the precise steps needed.
It also will not run under Mojave, so is quite a limited resource.