Elementary equivalence of profinite groups

There are many examples of non-isomorphic pairs of finitely generated abstract groups that are elementarily equivalent. We show that the situation in the category of profinite groups is different. If two finitely generated profinite groups are elementarily equivalent (as abstract groups), then they are isomorphic. The proof applies a result of Nikolov and Segal which in turn relies on the classification of the finite simple groups. Our result does not hold if the profinite groups are not finitely generated. We give concrete examples of non-isomorphic profinite groups which are elementarily equivalent.