The chords are all new. Can't speak for the rhythm patterns, but I suspect they're more or less similar. Nylon strings give a little more, which makes the split stroke and similar strums possible on ukelele, but maybe not on banjo.
The big difference is that you almost always use a pick on the tenor banjo, and hardly ever on uke, at least for strumming. Some folks do, of course, and when I play tenor banjo I often put the pick down use uke-syle thumb-and-finger strums. But a good tenor banjo responds best ot a plectrum, and a good uke responds best to bare fingers.
Often things that sound similar are done differently, like tremolo and very fast triplets. On the tenor banjo I think they snap the pick up-and down very fast, but on the uke most players use some kind of split strum. For instance, when I do fast triplets, I hit down with index, down again with thumb in the same stroke, and then up with one or more fingers. You don't have to move your hand nearly as fast as it sounds, and it impresses guitar players and other simple souls.
The tuning is supposed to be different, though lazy people like me tune them the same, whichever they learned first.
I agree with John- (Hi John, see you next month) Uke playing became popular at roughly the same time as Tenor banjo and many uke players play trad jazz. I also use fingers for uke and a pick for tenor banjo, but essentially do the same job, just on a different instrument. Of course, most tenor players play CGDA and ukulele is GCEA or ADF#B. I have played dixieland gigs on my banjo ukulele, with good results! Aaron