Let us quickly set aside their trivial attempts to subsume one another. Symbolists say nets are just an alternative hardware for doing computation. That may be one possibility, but it is certainly not the one that connectionists are exploiting when they use nets to learn. Connectionists say nets have the power of Turing machines. Perhaps, but not when they are used as nets, rather than as a hardware for implementing a Turing machine.
Symbolists say most nets are just symbolically simulated nets anyway.
True, but they could be implemented as real nets too. They and all their parts function in a way that supports a coherent semantic interpretation. They depend on the interpretations we make with our minds. And this kind of parasitism makes them ineligible as autonomous models of our minds, on pain of infinite regress. Consider an example.