Why do we multiply two wavefunctions in valence bond theory instead of adding them? The combined wavefunction of a hydrogen molecule in valence bond theory is the product of two wavefunctions, while in molecular orbital theory it is the sum of two wavefunctions.
Actually, the MO theory wavefunction can also be written as a product. While the individual molecular orbitals are just weighted sums of the atomic orbitals, the wavefunction can, to a first approximation, be written as a Slater determinant of the occupied orbitals. 

For your example of the hydrogen molecule, this would mean taking the antisymmetrized product of the occupied $\sigma$ spin orbitals. (Note, if you haven't encountered this before, spin orbitals are a spatial and spin description of a single electron, rather than the commonly encountered spatial orbitals which describe two electrons of opposite spin.