As Jess says, Schrödinger and Feynman are formally equivalent: either can be derived from the other. So if the question of which is more “fundamental” can be answered at all, it will have to be from other considerations. My own favorite way to think about the difference between the two pictures is in terms of computational complexity. The Schrödinger equation can be seen as telling us that quantum computers can be simulated by classical computers in exponential time: just write out the whole amplitude vector to reasonable precision, which takes exponentially many floating-point numbers, then update it step by step. The Feynman path integral can be seen as telling us that quantum computers can be simulated by classical computers in polynomial space: just add up the amplitudes of all paths leading to the quantum computer accepting, reusing the same memory from one path to another. Since polynomial space is contained in exponential time, the Feynman picture yields the better simulation—and on that basis, one could argue that it’s the more “fundamental” of the two representations.
As Jess says, Schrödinger and Feynman are formally equivalent: either can be derived from the other. So if the question of which is more “fundamental” can be answered at all, it will have to be from other considerations. My own favorite way to think about the difference between the two pictures is in terms of computational complexity. The Schrödinger equation can be seen as telling us that quantum computers can be simulated by classical computers in exponential time: just write out the whole amplitude vector to reasonable precision, which takes exponentially many floating-point numbers, then update it step by step. The Feynman path integral can be seen as telling us that quantum computers can be simulated by classical computers in polynomial space: just add up the amplitudes of all paths leading to the quantum computer accepting, reusing the same memory from one path to another. Since polynomial space is contained in exponential time, the Feynman picture yields the better simulation—and on that basis, one could argue that it’s the more “fundamental” of the two representations.