I think of Folk as a much wider umbrella of music, which encompasses songs tied to a certain culture or location with a large focus on story telling. And often has a very nostalgic feeling tied to that type of life.
Country I see as under the wider umbrella of Folk and mostly tied to Southern and Western music. There's obviously alot of great artists in Country who come from Canada, Australia, etc but they still keep that style of music. The roots are more based in Blues and Folk music from those general areas with alot of focus in songwriting on Working class life and Culture you'd see in the South and Southwest. And even though Country has some great song writers, I'd say there is more of an overall focus on Melody and Rhythm, as opposed to lyrics being primarily at the forefront of straight folk music.
Obviously Genre is very fluid and there are examples in both that don't fit with this at all, especially now. And there are many right ways to answer that question. But that's how I see the main difference, at least in stylistic origins.