Rabbinic Judaism and Christianity are the same age, though. They cover two ends of the propagation strategy spectrum. Paul tried to position Christianity as a pivot from Temple Judaism like the Sanhedrin did with Rabbinic Judaism, but the "light" version that spread like wildfire among Roman slaves is what survived. Roman slaves didn't really care about Jesus's messianic genealogy or all the allusions to him being the final pascal lamb, it was the salvation story ( itself a repackaging of exodus) that resonated. So Christianity sees itself as a successor while Rabbinic Judaism sees itself as a continuation.
Many religions explicitly require followers to try to 'spread the word' for exactly this reason.