Vicious Villians and Annoying Antagonists

During a discussion with an acquaintance about storytelling, the person always used the word villain when referring to the antagonist of the story. This rather bugged me, because I do not consider the words villain and antagonist to be synonymous.

According to dictionary.com, which of late I have elevated to the position of a minor deity...

villain: a character in a play, novel, or the like, who constitutes an important evil agency in the plot.

antagonist: the adversary of the hero or protagonist of a drama or other literary work.

The difference that I read into those two definitions is that the villain must therefore possess a quality of evilness and maliciousness, whereas an antagonist could just be someone who's getting in the way or working against the MC and friends.

According to my definitions, a villain can be an antagonist, however an antagonist does not have to be a villain. Sort of like how a rectangle can be a square, but a square doesn't necessarily have to be a rectangle.

In my head, that's the difference between, say, Severus Snape and Lord Voldemort. Lord Voldemort, I think we can all agree, is the villain of the tale. He's evil, wicked, and out to cause pain and destruction for others. Very not nice guy. Snape, I'm sorry, Professor Snape, on the other hand, is just out there to screw with Harry and make his life a littler harder. He's not evil or malicious. He just doesn't like the MC and is interested in making things a bit harder for MC and friends. I'd classify him among the antagonists.

My current WIPs do not include villains, just antagonists. People, especially in actual reality, tend not to be evil in nature (I'm not ruling it out, but it's statistically uncommon), and they also tend not to commit acts of evil for kicks and giggles (again, I'm generalizing, because sadly, some people like doing acts of evil for giggles. But I like to imagine they aren't part of the same species as I.)

However, the world has no shortage of people who are interested in getting in people's way. Well, life wouldn't be life without people being interested in crushing your dreams and making your day a little rainier, would it? (Normally, I'd say something about that being character building, but we all had a parent who said things like that. It probably came somewhere after stories about walking to school ten miles up hill both ways.) Antagonists, in my mind, are more common in the Other Side bunch.

Do you tend to have villains or just antagonists? Do you think villain and antagonist are the same thing?