The same holds true for Christ himself, but since his form is human, the omission allows less room for interpretation.
In the three films below, the differences between the Christs are subtle; the Satans are light years apart.
Some spoilers ahead, but no ruined twist endings.
On the surface of it, there's nothing overtly un-Christlike about him, and he even possesses a worn out, raggedy quality that can work well for the role. But I think if there's one main thing I expect from any visual depiction of Jesus, be it in fine art or in cinema, it's peacefulness. Not inhuman composure or stoicism, but some degree of peacefulness, even during the passion, even on the cross. Willem Dafoe, on the other hand, even in his most smiley-calm, looks like a man of a thousand demons. And considering the plot of Scrosese's film, that casting choice actually makes a lot of sense.
Satan is played by Rosalinda Celentano, though the pale, milky-eyed, androgynous character is qutie a long way off from the actress's real-life appearance. Satan's scenes were shot in slow motion; he does not blink, and his movements resemble floating or snakelike crawling rather than walking. Yet Gibson has intentionally chosen an actress whom he considered traditionally attractive for the role. “I believe the Devil is real, but I don't believe he shows up too often with horns and smoke and a forked tail,” Gibson said back in 2004 in an interview with Christianity Today. “Evil is alluring, attractive. It looks almost normal, almost good – but not quite.”
What I find particularly interesting in this film, other than the depiction of Satan (which, to Gibson's credit, is very different from any Satan I've ever seen in cinema), is the relationship between Satan and Christ.
Like in the gospels and the two films mentioned above, Satan tries to tempt Jesus by appealing to his human qualities – in the gospels, pride and greed; in Stevens' film, hunger; in Scorsese's, physical love – but in this film the chosen human quality is subtler. Here Satan focuses on doubt, appearing in Gethsemane to raise the question, “Do you really believe that one man can bear the full burden of sin?”
Another time he appears during Christ's flogging at the hands of the Roman soldiers, where he mocks Christ in a manner that appeals to his divinity rather than his humanity. Satan appears with what on first sight seems like a baby. He's wearing a hood reminiscent of Mary's veil, and is holding the supposed baby in a manner similar to the way Mary and the infant Christ are often portrayed in religious art. The baby is not a baby, and Satan isn't Mary; he's mocking Jesus by mimicking the symbols of the Christian faith as they will be hundreds of years into the future.
Satan, in that scene, is portrayed as the opposite of Christ and Mary – as an “anti-mother,” so to speak.
He appears in a few more scenes throughout the film, including as a manifestation of Judas's feelings of guilt, which drive him to suicide by hanging. None of those encounters are mentioned in the gospels.