Both words describe beliefs, opinions, or decisions that are strong and unchanging. Someone who has a firm belief in the importance of education is unlikely to change their mind. The word firm has a generally positive connotation—think of muscles, apples, and mattresses, but on the hardness scale it’s less than hard. Adamant, on the other hand, comes from the name of a legendarily hard stone. Adamant opposition and adamant belief are resolute, insistent, even immovable stances that will not give way despite urgings or appeals. When firmness reaches this degree, adamant is a strong synonym for firm. Both words can be used of a person: adamant about the curfew, and their associated words are very similar. Adamant can have a negative connotation and suggest rigidity if sticking to one’s guns becomes more important than the consequences.
Both words refer to imparting knowledge to someone. Teach suggests instruction in a subject or skill. Edify suggests instruction that benefits or improves someone morally, ethically, or spiritually. Neither teach nor edify is limited to an academic framework, however. We can say that an experience taught the boy how to stand in someone else’s shoes, and say that the experience edified him. Edify is often paired with verbs like uplift, inspire, and strengthen. Although the word is most heavily used in Christian religious texts, it has a definite place in general usage, especially in differentiating activities that do not strengthen or improve us from those that do. A book or movie that both entertains and edifies is something parents would urge on their children. After binge-watching a non-educational series, a person might wish they had done something edifying instead with their time.
Both words refer to the most important of a set of things. Main is very common, and can suggest not only the most important but also the biggest or most extensive: the main highway; the main reason. Main can attach to a plural—the main sources of energy, whereas paramount is only applied to the one thing that is of chief importance or concern, above all others: It is an issue of paramount importance; Safety is paramount. Paramount is a much stronger term than main, not only because it refers to a higher degree of importance, but also because it suggests something critical or essential that needs tending to.