If I am not mistaken, for a finite group $G$, $BG^+$ is equivalent to the product of the $p$-completions $BG\hat{_p}$. And by a theorem of Oliver, the $p$-completion of $BG$ is determined by the $p$-fusion system of $G$. The $p$-fusion system of $G$ is the category whose objects are subgroups of a $p$-Sylow subgroup $P$, and where morphisms are all homomorphisms that are induced by conjugation by an element of $G$.
So the question seems equivalent to whether non-isomorphic finite groups can have equivalent $p$-fusion systems for all primes $p$. I think there are such examples. I found the following one in a paper of Martino and Priddy (who attribute it to Minami): $Q_{4p}\times {\mathbb Z}/2$ and $D_{2p}\times {\mathbb Z}/4$. Here $Q_{4p}$ is the generalised Quaternion group of order $4p$, where $p$ is an odd prime.
Martino and Priddy give these as an example of two groups whose classifying spaces have stably equivalent homotopy types. But I think that they also provide an answer to your question in view of Oliver's theorem. The point is that they have isomorphic $p$-Sylow subgroups at all primes, namely ${\mathbb Z}/2\times {\mathbb Z}/4$ and ${\mathbb Z}/p$, and moreover the $p$-fusion systems are equivalent: you get the trivial structure at the prime $2$, and the only non-trivial morphism at $p$ the the inverse homomorphism.