If your question is really how would mathematics be different, the answer is — **it wouldn't**. Mathematics does not change based on which foundational theory gets chosen. It's the other way around, i.e., something is considered a foundational theory if it's able to express all of the mathematics we are interested in anyway.
Also, Solovay model is not a theory. It is a particular model of ZF showing the relative consistency of some statements independent from the axioms.
You are comparing apples and oranges, i.e., an extension of a theory (ZFC as an extension of ZF) with a particular model of that theory (Solovay model of ZF). so, the question about which one should be preferred is moot, we're talking about two different kinds of things.
If you insist on comparing the two, I would advise that you do not think of either ZFC or Solovay model as being the foundation of mathematics. Take ZF as the foundation and see both ZFC and Solovay as certain "specializations" of ZF, the former obtained by adding an extra axiom, and the latter literally being a particular realization (a model) of ZF.