"Firm" is a short, everyday word with a positive connotation. Someone "firm" has taken a good position and is going to stick to it, possibly because it is right, sensible, wise.
"He said no." "Can you get him to change his mind?" "No, it was a firm denial."
"She is a woman of firm convictions. She persists until she reaches her goal."
"Adamant" is much rarer, and more "literary." It means "firm" or "very firm" or "absolutely firm." (Literally, "adamant" is another form of the word "diamond," the hardest substance in the world).
"Stubborn" refers to someone who won't change their mind, never changes their mind, even when they are being wrong or stupid. "He's a very stubborn person, once he makes up his mind he won't listen to anything."
"Stubborn" can also be used about things that are difficult to move. "The lug nuts were rusty. I was able to loosen four of them, but the fifth was so stubborn I couldn't do it."
"Obstinate" means the same thing as "stubborn."
There is an informal, insulting term "pigheaded," which is even more negative than "stubborn."