There was a demo of Timeshift, before the developers revamped the game from the ground up.
Old demo: https://www.youtube.com/watch?v=vhufArkAAlQ
New demo: https://www.youtube.com/watch?v=mDfrO50iRKI

Jagged Alliance 2 had unique Demos prior to the release. I remember having played at least an english and a german Demo of the game. If I am not wrong, there also were some minor differences between the english and the german demo besides the language of course.

Strife: Quest for the Sigil had some different characters, different lines of dialogue, and the maps were altered for the demo. You were also required to complete the demo levels differently than what happens in the main game. From what I understand, Strife: Veteran Edition lets you choose to play the demo version and the full game, so kudos to them for that.

While I don’t have an answer to burkjon’s question, I have something that most of you will probably like: a way to find said demos ;)

Here you go:
https://archive.org/details/classicpcgames

Bulletstorm had Duty Calls

The demo for Operation: Body Count consisted of a level not featured in the full game, along with a weapon not present in it either.

You don't want to play Operation: Body Count though, Trust me on this.

The demo level to Bioshock was a little bit different than the starting level in the full version.

Suikoden II. The demo featured a dungeon that was not in the game.

That was actually the first proper FPS that I owned the full version of, and I think the only one for PC that I ever finished the full version of. Sure, it was generic -- but then, I didn't have a very broad frame of reference at the time. =P
The only difference I remember between the demo and the full game (not saying there were no other differences -- it's just literally the only difference I specifically recall!) was that the full game had some stupid device I thought of as the "punch gun" as the bottom-tier "last resort" weapon, whereas the demo had a shotgun (probably with infinite ammo) or something instead. I think.

As I remember the Age of Mythology demo was completely unique, it was a standalone campaign about some Norse Dwarves that didn't appear in the original game.

Age of Empires I had completely different campaigns
NOpe, it is a downloadable campaign, The Golden Gift
https://www.microsoft.com/en-us/download/details.aspx?id=22062

EDIT: My bad, golden gift has 2 dwarves which were in the campaing

RISE OF THE ROBOTS!!!

Cough *Just Like Oblivion* Cough.
I Loved Duty Calls It's a direct Ripoff of Call of Duty

This one was the first thing that came to mind when I read the thread title. I loved that shareware episode.

Does all the different Christams themed levels of games given out count?

It does a good job of warning the players how frustrating the game's design can get though. ;)

The main game has a better difficulty curve and is more fun than the demo, but it often applies the same design of locking the players in with powerful opponents without any options to rest or prepare.