How did lycans and vampires come to hate each other?

They don't. Real vampires and spiritual therianthropes are irritated with people who call themselves lycans, because 'Lycan' is a made-up term from the 'Underworld' movie series, and the term lycanthrope was never shortened in that fashion prior to the 'Underworld' film.

In the context OF that film, many spiritual therianthropes and vampires like lycans just fine.

Spiritual therianthropes also do not call themselves lycanthropes, even if they are wolves, because lycanthroopy is a psychological disorder in which a person is suffering from the delusion that they are turning into a wild animal. Spiritual therianthropes are not mentally ill.

Spiritual therianthropes do sometimes call themselves 'real weres', or 'real werewolves', etc. They do not, however, get furry...and they certainly do not turn into giant wall-climbing Mexican hairless dogs.