He is busy today, but he may come tomorrow. (it is possible but not certain that he will come, ability is implied)
Sorry, I overlooked your last question. Context is always important in posts and responses. So, another point, "John has not completed his school project and he may not go to practice today, but he may go tomorrow if he completes it tonight. (Permission). And occasionally "can" is used interchangeably with "may" for permission.
Student or Learner