I have built up a lot of experience with both these headphones, including the original Dx000 line of denons. But before I can give a proper suggestion, it would help to know of any other headphones that you've heard and what your impressions were of them, as the Beyers and Denons are really quite different from each other. That being said, I'll try outlining some of the differences between them that I found, and perhaps it will help you make up your own mind.
The DT880 are very clear, instrument separation/imaging is the best of the 3. The soundstage is decently wide (as wide as it has to be to give it's level of detail/separation, but not any more), but not as much as the D5000's. And while the DT880 has decent bass, it isn't particularly 'punchy' nor does it extend very far and it isn't compensated for a 'subwoofer-in-the-room' effect (important if this is what you consider 'neutral'). The highs aren't bad but they are slightly hyped, and the mids are just a tad bit recessed, but overall the sound is very balanced. I originally got this headphone for studio purposes, and while I like it and would say it's good for that, I wouldn't really recommend it for anyone looking for a 'fun' listening phone, unless you know that is really the type of listening experience you want.
The D5000 on the other hand, has rather neutral bass (taking into account 'subwoofer-in-the-room' compensation) if a tad bit hyped, and the the extension is great (it clearly goes all the way down to 20hz, and can rumble down to 1-5hz), but it is a bit muddy bass. The highs aren't bad, slightly warmer than the DT880's but still on the bright side, and the mids are recessed on these a good bit more than than the 880's. The soundstage on the D5000's is the largest of the 3, but the imaging isn't as clear as the 880s. I would also say that the D5000 is right on the edge of what I would call 'balanced' or somewhat 'neutral', even though they are colored as described above.
Then there's the D600's... these phones get a lot of flak, but I actually think they're quite good if you know what they excel at and adjust your expectations accordingly. Out of all the headphones I've ever heard, these have the best, clearest, most neutral compensated (i.e. like a studio subwoofer setup) bass, and are the 'punchiest' of the three. The extension is pretty much the same as the D5000's, but the roll-off begins at around 30hz, as opposed to 20-25hz in the D5000's, but the roll off is a lot more gradual than the D5000's, and it can still rumble down to 1-5hz as well (at normal listening volumes). The original Dx0000 line may appear to have more sub-bass at first, but that's just due to bloat/muddiness and harmonic distortion, whereas the D600's bass response is drastically cleaner/clearer with less distortion, hence I prefer them for that. BUT, and here's the kicker (the reason this phone is so 'controversial)'... the mids and treble are truly not very good. They were so disappointing when I first heard them, that I immediately began preparing to return them; luckily, I decided to give them another shot and burn them in for a good while, and in doing so, not only did the bass improve with that, but the highs/mids did too. Not nearly enough to actually be considered 'good' mind you (especially compared to the other two cans), but definitely tolerable. So I do like these headphones now, and use them to evaluate bass when doing studio work (only for mixing though, not mastering), as they really do have the best bass I have heard.
$340 (d600) vs $250 (dt880) also allows me to pick up the schiit magni/modi with the dt880 if I go that way.
