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 'supporter' and is used universally in the various Indian religions to refer to any doctrine that attempts to maintain life & the mind to be free from suffering & problems. The Thai teacher Bhikkhu Buddhadasa once explained "dhamma" as follows:
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, some religions will include the notion of "God" and practising dharma includes belief in God. Such beliefs in God help the believer reduce their suffering, such as believing when a loved one dies the loved one is dwelling in heaven or in the other world with God or Brahma.
In Pali Buddhism, dependent on context, the term 'dhamma' 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' specifically refers to the dharma of the Buddha, i.e., the Buddha's understanding & method to support life so life is free from suffering & problems. 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."
In summary, the words 'dharma' and 'Buddha-Dhamma' are not synonymous because Buddha-Dhamma refers to the Four Noble Truths, Dependent Origination, the Three Characteristics and Emptiness, which are not dhammas found in other Indian religions.