As a die-hard DC Bangai-O fan, I thought I'd throw in my two cents.
1) The game design complexity has been ramped up. The controls get a bit muddled in the process, but nothing that can't be taken care of with a little practice. I like the new layer of weapon selection before going into a stage.
2) The difficulty has been accordingly ramped up. I encountered at least a dozen stages that make the last level of the DC game look like a joke. I'm not a huge fan of the "shoot a line of exploding blocks and race to the finish" gameplay type or the "super precise flying or die" type. Also, Longai-o's can

right off. Bastards.
3) What the

happened to the story?! I felt that the quirky dialog of the DC game was half the experience...
4) Spirits is substantially more puzzle-y that action-y in comparison to the original. Not that it's a bad thing, just worth pointing out.
All in all, I definitely like Spirits as a sequel to Bangai-O. Still, I think the DC game will always be the best version in terms of design, feel, and atmosphere.
Now I have to go beat that infernal "Earth" stage... Argh.