The main differences between the two games is that in DoP you are out adventuring in the world for the sake of growing more powerful so that you can be victorious over the other covenants in town, or at least ally with them. What DoP has that DC doesn't is a clan vs clan mechanic and a diplomacy mechanic. Instead of being a lone adventurer, you are a member of a clan with a base and you are trying to make that base strong so that it doesn't fall to other clans in the town.
The combat is a bit more clunky in DoP, but in general the game is a bit deeper. There is still a lot of loot, and there is more to do with the loot because you can equip your AI controlled clan members. There are fewer classes in DoP, but you can do interesting things like buy monsters to guard your house and trade with other clans. In sum, DoP has a bit of a strategy game injected into it, whereas DC is a pure dungeon crawler. DoP also has an open world with occasional dungeons.