An antagonist works against the interests of the MC - they may not be a villain, may even be another hero as mentioned above, but they are at odds with the MC.
A villain is just a bad dude. Note that villains may not always be antagonists (I frequently have villains who work with the heroes for various reasons, at least for a while - and sometimes a villain is needed to "get the job done"), and antagonists are not always villains.
For example, in "Frankenstein: Or the Modern Prometheus" Victor Von Frankenstein is pretty much a villain - robs graves, abandons his child, runs away from his responsibilities after his fiancée is murdered on their wedding night, yet is pretty much the MC... The Creature becomes an antagonist after being rejected, and a villain after Victor destroys his second Creature instead of giving her life, but until the moment his Creator betrays him, The Creature is mostly innocent, and very naive.