What is the difference between a bulge and herniation?

None. When a disc protrudes from its normal site it is said to be bulging or herniated. These are essentially synonyms for the same pathology.
Degree of severity. A bulge is when the disc has lost some water content and has lost some of its ability to resist the pressure pushing down on it. A herniation is when some of the inner portion of the disc (imagine the texture of crabmeat) has pushed out and is pressing against a nerve.