A mashup is typically the cut and pasting of two different tracks together some times with other samples. The reason there are often Rap tracks mashed up with pop/rock tracks is due the fact that rap artists often release A Capella versions of their songs making them easily mashable.
Remixes usually involve the reworking of a track by manipulating the existing elements to extended them, make dub versions, etc. Most often today remixes are done in order to create crossover hits, such as Everything But The Girl did with "Missing" when Todd Terry was brought in to do the remix for it. In the case the remixer is creating an entirely new musical track to go along with the vocals. This time of remix is currently very much in vogue, with the likes of Bloc Party releasing entire albums of remixes.
Yes, this is true - but in order to "mash up" a song with another one, you often have to manipulate the existing elements in order for it to work or for the songs to fit together. For example, in that Jay-Z tune above where it is mashed up with Air, the tempo of the original Air track is slowed. Or take "A Stroke of Genius," if you want to hit on one of the early examples of mashups, where Christina Aguilera was paired with the Strokes. The Strokes tune was recut and rearranged to fit the Aguilera vocal track. Remixes often use outside samples of some kind, whether they be drum loops, keys, vocal samples, or whatever else a remixer would like to incorporate.
I guess my point is that the distinctions are nuanced; essentially, a mashup is just a more specialized form of remix, involving an additional song brought into the picture.
