I'm trying to learn more about grammar and often come across these terms. Not sure the differences.
Unhappily you are studying English at a time when the terminology - and the underlying understanding of how language works - is in process of change. Consequently, you cannot count on these terms to mean the same thing everywhere you encounter them.
Adverb phrase ... For some writers this means a phrase which acts as an adverb. For others the sense is restricted to phrases which are ‘headed’ by an adverb, in the sense that the adverb carries the core meaning and other words are in some sense dependent on the adverb, either modifying or complementing it. For the first group, both on January 15 and very rapidly would be adverb phrases; for the second group only very rapidly would be an adverb phrase, while on January 15 would be a preposition phrase.
Adverbial may be employed as an adjective, again in two senses. It may mean “pertaining to an adverb”, as when we speak of “the adverbial sense of fast” (as opposed to the adjectival sense). Or it may mean “having the character or role of an adverb”; linguists who employ adverb phrase in the second sense above would call a phrase of the first sort, like on January 15, an adverbial preposition phrase.
Adverbial may also be employed as a noun meaning “something which plays the role of an adverb”—for instance, an adverb phrase, in the first sense defined above, may be referred to as an adverbial.
Adjunct is less ambiguous: it signifies a word or phrase which does not play a required syntactic role but modifies a word or phrase or clause, providing supplementary information.
A group of words which functions as "an adverb", example:
He played the shot in a swift manner.
This is the same as "adverb phrase" but has wide area. Some words do not function as "adverb phrase" are included in an adverbial, example:
I went to him.
Every unnecessary and additional part which can be omitted without affecting the sense, example:
I am happy but you are sad.
I met your mother she was a kind enough person.
Difference between adverbial and adjunct
An adverbial consists of adverb equivalent words but an adjunct consists of all unnecessary parts of a sentence, example:
I pushed the boy (because he abused me yesterday.) = adverbial clause
I pushed the boy (and he began to abuse me).= adjunct