Merriam-Webster's Dictionary of Synonyms (1984) groups adamant with inflexible, inexorable, obdurate, and adamantine:
Inflexible, inexorable, obdurate, adamant, adamantine mean not to be moved from or changed in a predetermined course or purpose. All are applicable to persons, decisions, laws, and principles; otherwise, they vary in their applications. ... Adamant and adamantine usually imply extraordinary strength of will or impenetrability to temptation or entreaty [examples omitted]
The same dictionary groups tenacious with strong, stout, sturdy, stalwart, and tough:
strong, stout, sturdy, stalwart, tough, tenacious can all mean having or manifesting great power or force (as in acting or resisting). ... Tenacious comes very close to tough in its most general implications, but it places greater emphasis upon retentiveness of what has been gained or of adherence to a support, position, or idea; it carries a strong suggestion of holding on, of adhesiveness, or of maintaining strength or position in spite of all opposing forces that would dislodge , dispossess, thwart or weaken [examples omitted]When applied to material things and especially to substances it may suggest a powerful clinging quality and extraordinary resistance to forces that tend to effect separation or pulling apart [examples omitted]
So, according to this dictionary, the core difference in the two terms is that adamant applies centrally to resistance to being moved (physically or figuratively), while tenacious applies centrally to resistance to being separated from something (physically or figuratively).
S.I. Hayakawa, Choose the Right Word: A Modern Guide to Synonyms (1968) places the words in somewhat different families. Hayakawa puts adamant in a group with stubborn, headstrong, obdurate, obstinate, pertinacious, and pigheaded:
All of these words suggest a tendency to persist in an opinion, belief, decision, or course of action, generally with more force than reason. ... Adamant implies a hard and unyielding attitude that may be the result of a strongly felt or carefully thought out conviction regarding some important matter: [example omitted].
Hayakawa associates tenacious with willful, firm, hard-headed, hard-nosed, no-nonsense, strong-willed, and tough:
These words all describe more or less uncompromising or fixed states of mind. The differences between them reveal that obstinacy in itself can be adjudged either good or bad, depending on the motive behind it and the uses to which it is put.
... Tenacious, meaning tending to hold strongly, as opinions, rights, etc., is also much favored by men and women in the public eye: [example omitted]. Tenacious has the implication of hanging on, refusing to let go no matter what the odds against eventual victory. This can be interpreted as blind stubbornness or as fierce devotion to right principle. Nevertheless, the word always conveys some respect; a tenacious adversary, for instance, may be disliked but he is certainly not to be taken lightly.
Thus, although Hayakawa yokes the two words in question to some rather unsavory synonyms (adamant to pig-headed, for example, and tenacious to hard-headed), the book acknowledges that people sometimes use the words in an admiring way—as descriptive terms for someone motivated by “strongly felt or carefully thought out conviction regarding some important matter” (adamant) or by “fierce devotion to right principle” (tenacious).