How would someone describe the differences that make up the two?
Improvisation is spontaneously coming up with new music, or "composing in the moment".
A solo, on the other hand, typically means one performer playing a "crowd-pleasing" melody (or any passage) of some sorts, either completely alone, or with unobtrusive accompaniment from the others (paraphrasing Wikipedia).
A solo during a live performance is often improvised, but not always. For example, if a rock band plays a song off of an album of theirs, and the guitar solo is played exactly like on the recording, that is hardly improvisation. (Although perhaps it was improvised when the song was first recorded or composed.)
When a group of musicians is playing together, a solo is a portion of the song or concert that features one of the musicians in particular, be it the guitarist, the drummer, a sax player, etc. In a chorus setting, even a vocalist singing on his own is considered to be singing a solo. The solo can be a portion of a song, or it can be an entire portion of the program (is in, a Neal Peart drum solo). A solo need not be improvised: the soloists may know and have practiced the solo ahead of time, and may repeat the solo more-or-less the same way in every performance, or they may improvise portions of the solo, or the entire thing.
Improvisation refers to the practice of musicians playing music that has not been previously composed. This may or may not have anything to do with a solo. Yes, it may be that the soloist improvises her solo, meaning that when the portion of the song comes that features her, she composes her music on the spot. But this is hardly the only context for improvisation. For example, in a jazz combo, all the musicians may be improvising their parts, whether or not they are soloing; i.e. when the sax player is playing his solo, the pianist and bassist are improvising their respective accompaniment.
Even though when musicians improvise, they are making up the music right then, there still may be a predetermined structure that they understand and use to guide their improvisation. The jazz combo, for example, knows the chord structure of the tune ahead of time and so each member uses that structure to influence and inform the notes he chooses to play.
Of course, while he's improvising around that set of chords, the pianist may simultaneously be altering the chords or replacing some chords with substitute chords, while the bassist is still playing a bass line structured on the original chord changes. This can make for exciting music, but it also requires the improvising musicians to be listening to each other with great focus and reacting to each other appropriately.