Adverb is a word category, and adverb phrase (a phrase headed by an adverb) the corresponding phrase category. Adverbial is a function and may be realised by an AdvP (He spoke quickly), a PP (He spoke with enthusiasm), an NP (He's speaking this evening).
AdvPs do not always function as adjuncts: they may function as modifier in AdjPs (It was quite amazingly expensive), etc. Adverbial phrase is quite often used for any phrase functioning as adverbial and hence likely to be confused with adverb phrase.