After getting the plat for Demon's Souls and nearly getting the plat for Dark Souls... I felt that the breadth of Dark Souls was superior, but that Demon's Souls' design was much tighter with less room for bullshi.  The multiplayer in Demon's Souls was far better.  After putting in some effort into both, I could just never master the lag I suffered from connecting to the host in Dark Souls.

That being said, Dark Souls ultimately feels more open world.

Demon's Souls' design feels a bit more linear, but only in the sense that it gives you a few worlds to start in right after the tutorial.  You bounce back between these worlds (or levels) delving further and further in each time.  In comparison to Dark Souls, which ultimately dumps you right in with who knows how many ways to go (at first), it may seem limiting.  It's hard to say whether this is good or bad, because I love open world games.

As someone's already said, they're both really awesome games.  Your skills more or less transfer over into Dark Souls.  I actually felt a little bad for 360 players - jumping into Dark Souls for the first time must have been incredibly overwhelming.  Demon's Souls feeds you a little bit slower. 

that's like 15x as many 360s sold last week