The difference is small.
'Learn' means you don't know English. Or, perhaps you are still coming to understand English. To -study- is a general statement . As a native English speaker, I study my English to make a better statement. But, anyone can study English. Learners are studying also. Of course, in English, the meaning can change a bit depending on way the words are used, so this is not always true, but I am generally saying what it means