TITLE: Distinguishing pro-finite completions
QUESTION [12 upvotes]: Assume that we have two residually finite groups $G$ and $H$. Which properties of $G$ and $H$ could be used to show that their pro-finite (or pro-p) completions are different?
I asked a while ago in the group-pub mailing list whether finite presentability is such a property but Lubotzky pointed out that it is not the case. A finitely presented and an infinitely presented group can have isomorphic pro-finite completions.

REPLY [5 votes]: There's a theorem that two finitely generated residually finite groups have the same profinite completions if and only if they have the same finite quotients. A reference for the statement of this is Theorem 2 of this paper, but they cite Ribes and Zalesskii for the proof.