Dependent clause: A clause that cannot stand alone as a complete sentence. It needs an independent clause to make sense.  
Adverbial (Adverb) clause: A type of dependent clause that functions as an adverb. It modifies a verb, adjective, or another adverb.  

“Dependent clause” is a broad category.
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|–“Adverbial clause” is a specific type within that category.  

Here’s an analogy:

“Fruit” is like a dependent clause (a general term).
“Apple” is like an adverbial clause (a specific type of fruit).

Examples

Dependent clause: “that I saw yesterday” (This is a noun clause, acting as the object of the verb)

Adverbial clause: “because I was hungry” (This modifies the verb, explaining why something happened)  

Key features of adverbial clauses:

They start with a subordinating conjunction (because, when, if, although, etc.).  

They answer questions like how, why, when, where, to what extent.  

They modify verbs, adjectives, or adverbs.  

All adverbial clauses are dependent clauses, but not all dependent clauses are adverbial clauses.  

Here are some more examples to illustrate

Adverbial clause:

“When the bell rang, the students left.” (modifies the verb “left”)  

When = conjunction
the bell rang = clause
When + the bell rang = adverbial (adverb) clause

clause 2 = the students left.

adverbial clause “When the bell rang” modifies verb ‘left’.

Adverbial clause: “She was so tired that she fell asleep at her desk.” (modifies the adjective “tired”)  

Dependent clause (noun clause): “I don’t know [what he said].” (acts as the object of the verb “know”)

Dependent clause (relative “adjective” clause):

“[The dog (that chased the cat)] is mine.” (modifies the noun “dog”)