There is actually a (minor) difference. Amazon really is streaming the MP3s, while if you play one from Dropbox it downloads and plays and plays (technically it could be played while downloading, but this is actually uncommon).
There are slight technical differences - in streaming MP3s there is metadata embedded inside the files - see, for eg the second part of http://nicklothian.com/blog/2009/03/18/random-mp3-metadata-c...
Exactly why this would make a difference for licencing purposes I'm not sure, but I would note that there is a special "streaming" licence.