I know in the demo for Shadow of the Colossus on the PS2 you can access the garden area on top of the main shrine where you start. You can't do that in the actual released version.
Smash Brothers on the 3DS demo gives you up to 500 coins or something like that. I wish the playtime in the demo factored into the time locked characters. I used the one for Pokemon Alpha Sapphire, just sucks you can't use what you get till later in the game.
I remember the demo for Half-Life on the PC was actually a complete story unto itself called Half-Life: Uplink. Not exactly what you are asking for, but it does give you a reason to own the demo even if you already own the game.
I remember the Red Faction demo on my PS2 Jampack disc was completely different from the actual game, to the point that when I got the actual game I returned it because it wasn't nearly as fun as the demo.
But on topic, Bravely Default has been one of the best for that. The demo is it's own side story and it transfers some items, gave a nice game-start boost.