Essentially Both are Modifiers - which provide additional information.

Adverbial Clause always has a Subject and Verb and provides more information about the verb, adjective,adverb. It answers questions: How, when, where, why, to what extent, in what manner etc.

They start with a subordinating conjunction (e.g. because, when, although , provided that, as long as etc.) which joins otherwise independent clauses by creating relationships between them.

e.g. When the Monkey screamed, people got frightened.

In the above example When The monkey screamed is the adverbial clause. Here, The Monkey Screamed,an independent clause, combined with the subordinating clause 'When' is providing more information on why the people got frightened.

While, Adverbial phrases are of pattern Verbal + additional words (usually Noun/pronoun or another prepositional phrase). They function to create compact prose and variety/style in otherwise wordy sentences.

e.g.

  1. Looking lonely, Tom Walked into the room

The above example uses a present participial Phrase to economically express the two events which happened simultaneously.

Tom Walked in to the Room. He looked lonely.

The Phrase looking lonely doesn't have a subject - which can be determined by asking the question who looked lonely ?

Present Participial Phrases are used for events which happen simultaneously, Other tenses are possible as show below which can be used for different references of time :

  1. Past Participial : The audience, moved by the senators speech, clapped enthusiastically.

  2. Present Perfect : Having thought it over carefully , she decided not to apply.

P.S. Examples taken from

  1. Grammar Desk Reference by Gary Lutz and Diane Stevenson