When talking about the adjective versions of each word, it quite obvious what the difference is. When something is offbeat, its strange or unseemly. A search for the word yields synonyms such as “funny” and “weird.” Upbeat is peppy and cheerful! Ambiguity of either makes for some cheeky musical humor.
Musically the two words are actually related to their respective adjective forms. Offbeat is most often a criticism of accuracy. It’s a very broad musical term that literally means “off the beat”. You can refer to a specific one in some cases: the second sixteenth note in beat three would be considered an offbeat. But really, an offbeat can mean any part of the beat that’s not on the beat. This includes all of the wrong places to play a note. So when your music instructor says that you are offbeat, it’s probably not a good thing. Or even worse (or better for some of us), you’re just an odd creature who’s also inaccurate. But don’t take it too personally. Everyone’s offbeat in both ways at the same time at some point in their life. Just check out this trombonist’s sneeze. Offbeat and offbeat. It happens to the best of us. (Cheeky, yeah?)
Upbeat is more specific. An upbeat is always an offbeat, an offbeat is not necessarily an upbeat, an upbeat is never offbeat, and offbeats are rarely upbeat. Upbeat refers to a moment as far away from two downbeats as possible. Tap your foot. Clap. Tap again. Clap. Tap. You’re clapping upbeats (unless you’re offbeat). Sometimes upbeat can also refer to the imaginary offbeat an upbeat conductor gives to give his offbeat musical group to come in on the downbeat (or sometimes the next upbeat if the offbeat group is playing very upbeat music).
On that note, take a look at Upbeat Bird which is only as off the beat as you make it.