As I said in my previous post, the book I just finished reading (A Cappella Arranging) refers to the mashup as a subtype of medley. I disagree, thinking of them instead as two different styles that could be combined.
A mashup involves integrating two or more songs, such as by having parts of the songs sung together, or by mixing lines from the songs together to form new verses. A medley involves segments from two or more songs in succession, with some kind of transition in between.
For instance, this is a medley:
They move rapidly between songs, showing how the music of Beyonce has changed over the years.
On the other hand, this is a mashup (but not a medley, in my opinion):
Now, just because I think these are two different styles doesn't mean they can't be combined, and in fact, they often are. For instance, this Disney medley, by Voctave, featuring Kirstin Maldonado & Jeremy Michael Lewis (who are now engaged, by the way!), is mostly medley with a bit of mashup at the end:
On the other hand, the arrangement sung by the Barden Bellas in the final scene (composed by Deke Sharon), is a medley of three songs (Price Tag, Don't You Forget About Me, and Everything Tonight), with substantial mashing up (Just the Way You Are, Party in the USA, Turn the Beat Around - and according to Deke, there are two more songs in there I'm not hearing):
So medleys can vary in how much mashing up they include, from none to lots. However, that doesn't mean all mashups are medleys. The mashup I'm working on is more in the style of the "pure" mashup linked above, though it does spend more time with two songs in particular, with references to other two songs sprinkled in.