It depends on the context.

Sometimes they can mean almost the same thing, but as mentioned in other comments, "awesome" is more colloquial/informal. I would also say that "awesome" usually sounds a little bit stronger/more emphatic in meaning than "great." Ex: This movie is awesome! / This movie is great! (Same meaning, but "awesome" is stronger and more informal).

However, "great" can also mean something sort of like "excellent/admirable/high quality," and would be used in situations where "awesome" would not generally be used.

Ex: Beethoven was a great composer. Einstein had a great mind. The United States is a great country. Technically, you could use "awesome" in these sentences as well, but it would change the connotation and sound a bit silly.

they mean the same thing 

but awesome is used more by younger people and is more colloquial

awesome is slightly greater than great i.e. slightly more emphatic