DTS is an uncompressed format, whereas Dolby Digital 5.1 (or EX) IS compressed.
If you want a movie where you can tell a difference, check out Saving Private Ryan and Saving Private Ryan DTS. The clarity is AMAZING on the DTS track. DTS is usually what is used in the theatres, so if they have the DTS track they can slap it on the DVD without having to compress it. Unfortunately, because it's NOT compressed, it takes up a LOT of space so they usually skimp on the extras. Most of the time movies are released without the DTS track so they can add extras, then later on when they rerelease it they can charge you for the DTS track in a "special edition" package. Personally I just wish they'd put DTS on anything they have it for, fuck the extras. That's just me though.
If a DD 5.1 track is done right, you can barely tell a difference.
Regular Dolby Digital (not DD 5.1 or DD EX) is only a Pro Logic surround, where the back to speakers are counted as ONE channel and the sounds are highly compressed. Sometimes it's hard to make out the rear channel because of this.
Most Anime DVDs (TV series, not movies and the like) at just DD (Pro Logic that is). All DVDs use Dolby Digital, whether it's 5.1, Pro Logic or DD EX. DD Ex adds a center channel to the rear speakers, thus you get 6.1. DTS EX is the same thing (6.1)
If given a choice between a DD 5.1 track on a DVD with extras, or DTS without extras, I go for the DTS. I barely watch extras anyway.