The cow in Apocalypse Now

TL;DR: Yes, the animal was really killed in Apocalypse Now, and although the slaughter wasn't illegal, it did have some repercussions for the film.

It really happened:

The animal (a water buffalo, or carabao) was killed - but not for the film. The tribe in the film was a real indigenous tribe that lived in the area, and they had already decided to slaughter it. Coppola merely decided to film the event.

The buffalo was already marked for ritual sacrifice by the indigenous tribe cast as the disciples of Brando’s gone-native Colonel Kurtz, and the argument was that it would have been hacked to death whether or not the cameras were rolling.
- AV Club

It wasn't illegal in the Philippines, where the scene was filmed:

I'm not familiar with animal welfare laws in the jungles of the Philippines in the late 1970's, but the water buffalo belonged to the tribe, and this is simply how they slaughtered their livestock. By all indications, there were no laws preventing them from doing so, nor was it illegal for Coppola to film them doing it.

Even today, it seems that an indigenous tribe like the Ifugaos would be able to conduct a similar slaughter of carabao:

SECTION 6. It shall be unlawful for any person to torture any animal, to neglect to provide adequate care, sustenance or shelter, or maltreat any animals or to subject any dog or horse to dogfights or horse fights, kill or cause or procure to be tortured or deprived of adequate care sustenance or shelter, or maltreat or use the same in research or experiments not expressly authorized by the Committee on Animal welfare.

The killing of any animal other than cattle, pigs, goats, sheep, poultry, rabbits, carabao, horse, deer and crocodiles is likewise hereby declared unlawful except in the following instances:

1. When it is done as part of the religious rituals of an established religion or sect or ritual required by ethnic custom of indigenous cultural communities: however, leaders shall keep records in cooperation with the Committee on Animal Welfare...
- Republic Act No. 8485: An Act To Promote Animal Welfare In The Philippines, Otherwise Known As “the Animal Welfare Act Of 1998”¹

However, the Wikipedia article for the movie says that it may have been illegal under U.S. law, because the film was an American production:

Although this was an American production subject to American animal cruelty laws, scenes like this filmed in the Philippines were not policed or monitored.
- Wikipedia

It still had consequences for the film:

Because of this scene, the film was not given the familiar "No animals were harmed in the making of this film" credit.

The sacrifice of a carabao, which takes place during the Kurtz episode, was "a real ritual slaughter performed by the Ifugaos." As a result, Apoca- lypse Now is one of the few mainstream Hollywood films not to carry a statement in the closing credits that no animal was harmed during the making of the picture.
- Godfather: The Intimate Francis Ford Coppola, by Gene D. Phillips

The American Humane Association has added Apocalypse Now to its list of "Unacceptable" films.

Featured Animal Action
Apocalypse Now was filmed in the Philippines in 1979. According to AHA's research, a water buffalo was hacked to pieces during the making of the film, earning the film an Unacceptable rating from AHA.
- AHA

Notes:

¹Thanks to Ixrec for finding this.