The most obvious difference between the games is the combat system. DD featured real time / action RPG combat, whereas D: OS will have turn-based combat with action points (similar to Shadowrun Returns, X-com:Enemy Unknown, etc.). IMHO this makes D:OS so much better than DD. I'm a big fan of turn-based tactical combat... Hate anything that boils down to button mashing or in the case of DD (hold down the button). I recently tried Divine Divinity for the first time and quit after awhile, largely due to finding the combat so boring.
The other big (also obvious) difference is that D:OS is graphically superior to DD. It will support modern resolutions (widescreen aspect ratios) whereas if you play DD on a modern monitor you're stuck with large black borders and a smaller playing space. D:OS is still isometric, but it looks so much better.
Where the similarities come in, I think, will be that both games have similar quest design. Larian prides themselves on giving the player choices when it comes to how you go about completing a quest. And of course it takes place in the same world with the same lore, etc. though D:OS is actually a prequel.