They made Dark Souls more difficult. In Demon's Souls you could kill practically every boss (including the False King) by shooting from a safe distance. Bow was like a god-mode cheat. It makes boss fights easy or at least as easy as you want them to be. Dark Souls has much more challenging bosses because you can't just shoot them with hundreds of arrows. It is somewhat alleviated it by giving an option of in-game companions, but still a lot of boss fights are rather tough.
There are some noticeable differences in the design. Dark Souls has a single connected world, with some in-game transitions, Demon's Souls has separate location accessible through portals in Nexus. Demon's Souls has a unique feature - worlds get white (easier, but less souls) or dark (harder, more souls) depending on bosses killed and protagonist dying, which in turn provides unique items and enemies.
I would say if you can, go play them both. I bough my ps3 just to play Demon's Souls (+ Heavy Rain, God of War and Uncharted series, + The Last of Us and BTS now), and it was worth it.