What's the difference between a verb phrase and a predicate?
In the wider definition, there is no difference; they are the same. In a narrower definition also used, a verb phrase is only part of the predicate-- it includes only the direct verbal forms (auxiliaries, participles, etc):

I have been skiing all day.

I -- Subject
have been skiing all day -- Predicate (and Verb Phrase in the wider sense)
have been skiiing -- Verb Phrase (in the narrower sense)
all day -- Adverbial
All predicates are a verb phrase, but the reverse is not true.
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 Mister Micawber's reply was promoted to an answer.
Alright... that is fine.
These two terms fall into two different categories: predicate is a part of sentence and verb or verb phrase is a part of speech.

Ania Kieturakis
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There is no spesific difference of them

But we can analyze, the verb pharase is original verb based on to be and so on. In predicate have the adverbial or complement in sentence