I'd say DMC1 is better than DMC3 in every regard EXCEPT gameplay. When I want a more thrilling gameplay experience, I play DMC3/DMC4. When i want better atmosphere, characters, locations, etc., I play DMC1. I just love the atmosphere and overall feel of DMC1... The music was awesome, the bosses were cool, the enemy designs were fantastic...and yeah, Nelo Angelo FTW!
The only thing that gets me is the difference in controls between the two games. I've played DMC3 and DMC4 so much that I can barely play DMC1 now. However, it is still a great game, and I feel it always will be.
EDIT: Well, DMC3 has a more in-depth story than DMC1 had, and Vergil was really awesome... I guess story-wise and action-wise, DMC3 would be better. But as far as general atmosphere goes I'm still going with DMC1 in that regard.