I would focus on just those individuals captured by either the hospital or the provider but not both, and then look for differences beteen the two populations using the usual two-sample tests, such as a t-test.
As an example, consider a binary characteristic, like sex. Suppose the data were like this:
male female total percent male
prov only 58 42 100 58
hosp only 83 117 200 42
both 355 345 700 51
If we're going to compare the percent males captured by providers and hospitals, we could compare (58+355)/(100+700) to (83+355)/(200+700), but the 355/700 are in common, so I suggest just compare 58/100 to 83/200 (say, using a $\chi^2$ test). Alternatively, as suggested by whuber below, you could compare the three groups.
If the outcome is some continuous number, I'd replace the $\chi^2$ test with a t-test or ANOVA.