The difference between adjective complements and adjective phrases depends on the difference between the parts of a clause and the parts of speech.
We use the phrase "parts of speech" to refer to the nature of words. Parts of speech include things like nouns, prepositions, adverbs and adjectives. We can apply the same labels to units that are larger than a single word -- meaning that noun phrases, prepositional phrases*, adverb phrases and adjective phrases are all labels that refer to collections of words that act as a single grammatical unit and that fulfill the same functions as the individual words. An adjective phrase is a phrase that contains an adjective and that can do the same job as an adjective on its own.
We use the phrase "parts of a clause" (or, more traditionally, "parts of a sentence") to refer to the ways that words and phrases relate to each other -- to the jobs that words and phrases have. Parts of a clause include things like subjects, direct & indirect objects, and complements.
An adjective complement is an adjective (or an adjective phrase) that fulfills the job of a complement.
Consider the following:
He is a very happy man.
That man is very happy.
His hobbies make him very happy.
In every one of the sentences above, the phrase "very happy" is an adjective phrase. The adverb "very" modifies the adjective "happy", and the two words together can do the same jobs that the adjective "happy" can do on its own.
In the first sentence, "very happy" directly modifies the noun "man", and it is part of the noun phrase "a very happy man". In turn, "a very happy man" is a part of the clause, specifically a predicate nominative subject complement.
In the second sentence, "very happy" has a different job. Instead of being a part of a noun phrase, it is the part of the clause that we call a predicate adjective subject complement.
In the third sentence, we can see another example of an adjective complement. This time, "very happy" serves as a predicate adjective object complement.
The label "adjective complement" applies to both predicate adjective subject complements and predicate adjective object complements. It applies to both one-word adjectives and to adjective phrases -- no matter how long or involved the adjective phrase may be.
When we talk about adjective phrases, we're talking about the nature of the phrase or the way the phrase is built. When we talk about complements, we're talking about the way that something fits into its clause or the job that some word or phrase satisfies.
When we talk about adjective complements, we're mentioning both the nature of the word or phrase and the kind of job it performs.
* Prepositional phrases are a special case. Prepositional phrases consist of prepositions and their arguments. Unless we expand this discussion to include intransitive prepositions, it doesn't make much sense to say that prepositional phrases do the same jobs as prepositions on their own. For a similar reason, we can avoid the complexities and confusion of the label "verb phrase".