I think it's worth mentioning here that a major reason that guitarists and other stringed instruments use an alternate tuning is to accommodate a singer's voice. Though I believe you are referring specifically to instrumental music given your example, this fact also supports the case that D# tuning (usually called Eb tuning) is not that rare at all, at least in pop-music. It's fairly common knowledge that artists like Jimi Hendrix, Black Sabbath, and Led Zeppelin used Eb tuning.
As previously stated, it is common to use an alternate guitar tuning to create a heavier/darker mood and/or increase the ability to bend the strings (by tuning lower than standard.) However, I have also heard the virtuoso instrumentalists, like Steve Vai and Joe Satriani, sometimes tune to a semitone higher or lower for the explicit purpose of making their riffs and licks more difficult to understand and replicate.
And considering that the determining the "best" of anything is highly subjective in nature, I'd have to agree with the other answers in that it is all about what you want to get out of the music. Using an alternate tuning exposes you to a different spread of sound which could potentially even contribute to inspiring a new song or three...
To expound on leftaroundabout's answer, another alternate bass tuning, for 5 and 6 string basses, is to drop to Bb on the low string; essentially used for the same reasons as dropping down to Eb. In my own experience, this can come in handy when accompanying and/or replicating an 808 in hip-hop or reggae. But, dropping the low B any more than that tends to make the sound way too muddy.
EDIT
See this question and this question for similar inquiries and responses...