Are the words Dhamma and Buddha Dhamma synonyms? Or are there hidden differences between between both? Are these words used in different circumstances?
Th word 'dharma' or 'dhamma' means 'that which supports' and is used universally in India.
The term 'dhamma' in Pali Buddhism is translated in different ways such as 'phenomena', 'nature', 'law', 'truth', 'path', 'skilful practices', 'fruit', etc, because each of these are things that 'support' the spiritual life.
The term 'Buddha-Dhamma' refers to the dharma of the Buddha, i.e., the Buddha's understanding & method to support life so life is free from suffering & problems.
Dhamma is the thing called "the Duty of all living things" that which they must do to survive both physically and mentally, both for their own sake and for that of society. Even when translating this word as "teaching," "learning" or "practice," the important understanding is still in its being the duty of salvation. Whenever duty is done, that is Dhamma practice.
For example, when Sariputta (Upatissa) first met a Buddhist monk:
Upatissa said: "Serene are your features, friend. Pure and bright is your complexion. Under whom, friend, have you gone forth as an ascetic? Who is your teacher and whose dhamma do you profess?"
Assaji replied: "There is, O friend, the Great Recluse, the scion of the Sakyas, who has gone forth from the Sakya clan. Under that Blessed One I have gone forth. That Blessed One is my teacher and it is his Dhamma that I profess."