Starcraft: Stracraft Playable Demo, later available as Precursor Campaign (prequel campaign)
Warcraft III: Reign of Chaos: Warcraft III: Reign of Chaos Playable Demo (prequel campaign)
Fallout: Fallout demo (different story)
Half-Life: Half-Life: Uplink (cut content)
Age of Empires: Age of Empires Trial Version (additional campaign and scenario, can be copied to the retail version)
Settlers IV: Smack a Thief ("techdemo" minigame)

- these are the ones I know.

EDIT 2015: http://www.abandonia.com/vbullet/showthread.php?t=23560

Not a prequel, but the Freddy Pharkas demo has puzzles not in the game.

I remember the Glider 4.0 demo had an interesting set of levels not found on the full release.

'Jack in the Dark' was a single-floppy prequel to Alone in the Dark 2 - you played as the girl who is lost in the AITD2 who is locked in a toy store. It came free with AITD2 and with later cd-rom editions of AITD.

The Wing Commander 3, 4 and Prophecy demos are all unique.
http://www.wcnews.com/news/2006/03/31/the-cic … is-a-demo-cracy

And so is the Starlancer demo:

The trial version contains two never before seen missions and its own unique carrier ship, the Yamato. These intense missions are actually the prequel to the StarLancer you will see in stores, revealing the events that led to the transfer of Commander Enriquez to the Reliant and her subsequent command of the rag-tag 45th Volunteer Squadron.

The Space Quest 6 demo.

I never succeeded in making that Fallout demo to work. I didn't try with more recent versions of DOSBox though