IMO, 見ます means "to see (a thing)", and 見えます means "to be visible". 見ます takes an object (transitive) while 見えます doesn't (intransitive).
映画を見ます。
I see (watch) a movie.
小野さんは若く見えますね。
Ono-san looks young. (Ono-san is visibly young)
10キロから、富士山を見る。
From 10 km, I see Mount Fuji.
10キロから、富士山は見えます。
From 10 km, Mount Fuji is visible (Sometimes translated as "I can see Mount Fuji")