The 1998 movie is called Godzilla. The monster in the movie is called Godzilla too. The monster was featured in the television cartoon Godzilla: The Series, still called Godzilla.
The monster later appeared alongside the real Godzilla in Godzilla: Final Wars, this time with the trademarked name of Zilla, as shown in the end credits. The name change reflected the popular and long-held notion that the monster was not, in fact, Godzilla.
Since then, the monster has appeared in the Godzilla: Rulers of Earth comic book from IDW, still called Zilla.
It's just a giant lizard character named Zilla. It used to be named Godzilla. Now its name is Zilla.
The monster in the 1998 movie is still called Godzilla, not Zilla, just as the butchered Led Zeppelin recording that plays over the end credits is rapped by a performer called Puff Daddy, not Diddy. Sometimes names change but that doesn't mean the past is altered.