The "multiple universes" theory is an explanation that if quantum mechanics describes both the quantum mechanical system under study and the observer, then after the interaction of "observation", the system looks like a superposition of independent systems that each look like the observer observed something different. The amplitude of each system corresponds directly to the likelihood that we will think we observed one thing versus another, and quantum mechanics + thermodynamics shows that these copies will never be able to have any meaningful interaction.
In short, in this interpretation reality, is quantum mechanics. There is never a "collapse" as there is in the Copenhagen interpretation. "Multiple universes" is merely a description of what quantum mechanics predicts the outcome will act like.
If you fire a particle at two slits with a screen on the other side you get an interference pattern on the screen implying the particle in a sense went through both slits and the results interfered. This happens even if you use quite complex particles such as C60 buckyballs. So the universe in some sense runs multiple versions some with the molecules going one way and some going the other. All the "multiple universe" explanation implies really is that this is a general phenomena and the universe runs all possible versions even when we can't observe the results.
Both of them predict correctly because they both actually use quantization of particle position. The issue with both of them is that both interpret that solution to equations as a real object - QM takes it for particle itself while pilot-wave theory assigns some real world existence to that "Pilot Wave". It would be like in case of shooting a bullet from a rifle QM would say that the trajectory is the bullet, where is pilot-wave would say that there is real trajectory object guiding the bullet.