Fürst is a title of a ruler. Prinz means that hes following in the line of ruling.
Fürst is a lower title than König. So a Fürst might rule over a smaller territory and might also be a vasall. Its a title similar to duke (Herzog) or count (Graf). Both those titles are below the Fürst and so also below the König.
In addition to what MortiBiRD said:
I know it's confusing because in English there's (1) "prince" = Fürst (noble rank), ruler in a broader sense; and (2) what apparently they call "prince of the blood" = Prinz, son of a king (or, in German, of a Fürst (some of them anyway) or a Graf), also used for fairytale princes.
As for "prince = Fürst", they share the same root: Latin "princeps", which means something like "the foremost one". An sort of translation during the early middle ages was "forista", from which derived "Fürst", which is thus related with English "the first".