Instead of Dark Souls scattered campires, you get one safe hub world where you can return to between "levels".
Also, instead of one big open world, the game basically has 5 or so huge "stages" split into 4 aforementioned levels each(which are also quite large) with a boss at the end of each afterwhich you get the checkpoint you can return to the hub world and spend your souls.
One one end, Demon Souls can be easier because magic can regen with an item you get by playing the "royal" class- basically allowing you to cheese enemies with projectile spells at the start.
On the other end Demon Souls is tougher because its much less generous with its access to the save points where you can spend your souls (you basically have to clear entire stages in one run before you can spend souls-die at the boss? yes, you must go through the whole stage again).
Also you are highly gimped when dead with half of your life bar, where hollow mode was really no big deal in Dark Souls.