There is indeed a difference in meaning.
The dative is used to introduce the existence of something and assign it to a referent. Nomen mihi Adam est is a merger of two underlying ideas: I have a name, and that name is Adam. If you say Meum nomen Adam est, you are stressing who the name belongs to. In other words, "I am the one with the name of Adam" as opposed to other people. The word nomen in this case is already assumed to be part of the conversational landscape.
This is how Allen and Grenough expresses the difference:
- The Dative is used with esse and similar words to denote Possession:— “est mihi domī pater ” (Ecl. 3.33) , I have a father at
home (there is to me). “ hominī cum deō similitūdō est ” (Legg. 1.25)
, man has a likeness to God. quibus opēs nūllae sunt (Sall. Cat. 37),
[those] who have no wealth. [*] Note.--The Genitive or a Possessive
with esse emphasizes the possessor; the Dative, the fact of
possession: as,—liber est meus, the book is MINE (and no one's else):
est mihi liber, I HAVE a book (among other things).