I think it could be argued that it is Mr. Utterson who is the protagonist of "The Strange Case of Dr. Jekyll and Mr. Hyde." Even by using the word "case" in the title, Stevenson suggests that this narrative is somewhat like an official or legal record of what Utterson observed and learned about the activities of Dr. Jekyll in connection with the experiments that led to the doctor's changing into a different character given the name of Mr. Hyde. Stevenson takes great pains to introduce Mr. Utterson at the very beginning, describing him in detail. Much of what happens throughout the tale is narrated through Utterson's point of view. Most importantly, Utterson has a strong motivation to discover the truth about what is going on in the life of Dr. Jekyll, who is his old friend and also his client. Utterson wants to find out who this Mr. Hyde really is and what sort of a hold he has over Henry Jekyll. If Mr. Utterson is the protagonist, then Mr. Hyde is the antagonist. It is very significant that early in the tale there is the following decision made by Mr. Utterson, a decision which guides him to the very end:
"If he be Mr. Hyde," he had thought, "I shall be Mr. Seek."
Good question. They are two different men and one in the same man. A protagonist is a central character to the story, the one who drives it. An antagonist is the character/force working against him.
In this story, Dr. Jekyll is the protagonist. He is a man trying desperately to control the baser instincts of his personality - the instincts of passion, anger, etc., that can cause a human being to be irrational. He has lofty and idealistic goals. He does this by splitting his personality into two so that one aspect can be entirely focused on knowledge and learning - and, therefore, be more productive - and the other focused on the pleasure principle only.
His antagonist is the other half of his personality, the side that is animalistic and is known by the name Mr. Hyde. This "man"/split personality wants... and works to get what he wants. He wants to be in control all of the time, and he works against Jekyll to do that. So, really, Jekyll's antagonist is himself. He is in conflict with his passion and desire.