A good and simple answer to my question. I'd forgotten about the proselytizing and universality angle. Christianity has an in-built narrative about the Jews: those are the people who didn't recognize their own Messiah. But this is treated as a thing of the past.

Islam on the other hand makes an open challenge to Christianity. (From a Judaic perspective, it doesn't matter what Christians believe, because Christians are gentiles, whereas in Judaism the only people of interest are Jews.) It claims to correct alleged errors in the New Testament. To quote from the New Yorker article about the new Pope (not online yet) that prompted this diary:

It should be remembered that John of Damascus, the eight-century saint and last Father of the Church, considered Islam to be a Christian heresy; today, by strict Catholic definition, any religion that postdates and rejects the divinity of Christ is heretical

This point raises a further reason why Islam is more problematic for Christianity than Judaism: Islam postdates Christianity, whereas Judaism predates it.

A bomb, H bomb, Minuteman / The names get more attractive / The decisions are made by NATO / The press call it British opinion -- The Three Johns