Neptune plays a central part in The SpongeBob SquarePants Movie. The movie also introduces Neptune's daughter, Princess Mindy. This version of Neptune is green-skinned and far more cartoony than the version in "Neptune's Spatula." He is also explicitly portrayed as a king rather than a god, and lives in a palace just outside of Bikini Bottom rather than Atlantis. This Neptune is very brutish and tyrannical, which he tells his daughter is necessary for him to be respected as a ruler. His crown is very important to him, not only because it is a symbol of his power, but because it covers his "thinning" (bald) head. When the crown is stolen by Plankton, Neptune attempts to execute Mr. Krabs, who is framed for the crime.
Judging by his different appearances, it is reasonable conclude that Neptune is actually two different characters, with the first one referring to the Roman deity of the same name, and the second one works as the actual King of SpongeBob's underwater world. Their different places of living and the fact that he doesn't know SpongeBob and Princess Mindy only being in the movie reinforces this theory.
Another possible explanation is that he has gotten older from the show to the movie resulting in his loss of hair, because Stephen Hillenburg did establish that the movie does take place after the series.