Broadly speaking, Feminism does mean to oppose sexism and level the field between men and women. There are of course going to be people, overwhelmingly men, who are opposed to that in the first place, but we don't care about their opinions right now.
Unfortunately, some people have taken to using the label of Feminism not to advocate for women, but to attack men. This comes in both the form of denigrating men as a whole and also attempting to erode their rights specifically. I don't think it's an especially large group, but rather is fairly vocal by nature and their message gets amplified by men eager to feel like they're actually the victims. Either way, it does somewhat taint the well of "Feminism" when a portion of it's purpose seems to just flip the oppression, rather than eliminate it.
Anti-sexism then is a more neutral term that doesn't have that "Us vs Them" mentality baked into it. It also has the added benefit of including advocating for those who don't strictly define themselves as Men or Women, being a term that opposed sex-based discrimination of all forms. Ideally, Feminism and anti-sexism would be basically synonymous, and in most use cases it would be understood to be. When presented side by side, though, the finer details will stand out more.