Many times they are the same.
There is a slight difference.
A person might persuade or convince another person to do or believe something.
Something (not a person) might convince a person (not persuade).
He persuaded me to go on the trip with him. (Could have said "convinced")
After seeing the snow on the roads, I was convinced of my choice not to drive.
Also, ''convince' has a slightly (not much) more positive sense associated with it than 'persuade'.