I happen to own both the HP2 and the DT880s, but besides both being wonderful headphones and great sounding I don't think they sound all that similar...but in fact quite complementary.
Head/soundstaging are probably the biggest differences. The HP2s, while more forward and with a smaller headstage, has a great sense of permeability that make it seem very open otherwise. Conversely the DT880s have a great soundstage with instrumentation that is wide and deep, but at the same time there seems to be a certain boundary that encloses the whole performance for a sense of intimacy.
The HP2 has a tremendous amount of neutrality and stoicism, with a sound spectrum that seems to be flat from top to bottom without any particular emphasis. The DT880s meanwhile has a very pleasant but somewhat rounded bass and slight treble emphasis which adds a nice sparkle and a perceived sense of detail. However, if you listen more closely it is the HP2 which has much better detail, resolution, and delineation of the notes. As monitors, the HP2 can be perceived as analytical but it doesn't suffer the brightness, dryness, or sterility that are usually associated with such equipment...it's just very natural and presents music without its own coloration.
As others have also mentioned, the DT880s are warm sounding but in a way a bit thinner sounding than the HP2, which has a substantial amount of weight and mass behind the sound.
I find both headphones very enjoyable for different reasons. I turn to the DT880s when I want musicality, a wonderful sense of space, and when I want to be moved by the music. I use the HP2 when I want neutrality, accuracy, and when I want revelation into the music.