Full Idea
A weak completeness theorem shows that a sentence is provable whenever it is valid; a strong theorem, that a sentence is provable from a set of sentences whenever it is a logical consequence of the set.
Gist of Idea
Weak completeness: if it is valid, it is provable. Strong: it is provable from a set of sentences
Source
George Boolos (On Second-Order Logic [1975], p.52)
Book Reference
Boolos,George: 'Logic, Logic and Logic' [Harvard 1999], p.525
A Reaction
So the weak version says |- φ → |= φ, and the strong versions says Γ |- φ → Γ |= φ. Presumably it is stronger if it can specify the source of the inference.