Hey Josh,
Let me open with a disclaimer:
I've never used a
DR-880 and I don't even know what a Delta 1010 looks like. I'm only responding because I have a DR-770 (which is pretty similar, I think) and because you haven't gotten many replies so far. I should also confess that I haven't yet used my DR-770 very much yet. That said, here's what I can offer:
The DR-770 allows the user to send out each drum voice to a separate track, or to group several voices together, or to send the whole performance to the recorder panned through stereo outputs. Unfortunately, in addition to the set of stereo outputs the 770 only has a single individual output. Therefore, in order to record separate voices to individual tracks, you would have to use midi to keep all the voices in synch with one another. I set up my machines so that the recorder is the master and the DR-770 is the slave, and then I start laying down the tracks.
bluesky's right that many drum machines allow you to do some basic EQing and other processing before you send the signal to the recorder, but unless the 880 is a lot more sophisticated than the 770, you'll only be able to process the overall output, not each separate voice. And it isn't very likely that you'll want to apply the same EQ settings to the kick that you apply to the snare. Besides, if you have the ability to track voices individually, you might as well do it. Then you can tweak to your heart's content, process the hell out whatever you like, and be doing it while listening to the whole mix instead of just the drums.
One last thing: If you find the cymbal voices on your machine as objectionable as most people do, you should at least record the cymbals onto their own track or tracks so that you can overdub them with real cymbals later if you should get the opportunity.
Hope this helps.
HC