How is many worlds different to basic probabilities? In the many worlds theory the universes branch according to the wave function. We find ourselves travelling down one path of an immense number of these branches of a tree. When we branch, we have no interaction with the other branches.
This sounds to me very similar to classical mechanics where we have a probability function which defines the chances of a particular outcome to occur. When we observe it we see that in this universe we got a particular outcome. But there are also nearly an infinite number of other imaginary universes where the other outcomes occurred according to the probability function (similar to the wave function).
I guess I am asking what’s the difference between the imaginary universes of non-realised outcomes, and the many worlds branches we never travel down? We have no interaction with them so they are as real as our imagination.
 A: 
In the many worlds theory the universes branch according to the wave function.

I would consider this to be not quite right.
Many-worlds is not a theory, it's an interpretation (MWI). For all practical purposes, in essentially every experiment ever done, its predictions are the same as those of, for example, the Copenhagen interpretation (CI). Therefore they are not different theories.
MWI also doesn't have to involve the concept of branching. The talk about branching is more of a heuristic or a way it's presented in popularizations. The most austere versions of MWI simply posit the same postulates that everyone agrees on for quantum mechanics, and doesn't add an extra postulate about collapse as in CI. No branching.

I guess I am asking what’s the difference between the imaginary universes of non-realised outcomes, and the many worlds branches we never travel down? We have no interaction with them so they are as real as our imagination.

(1) Classical probability doesn't allow for interference. Therefore the ways in which we predict probabilities are different in quantum mechanics than in a classical stochastic system.
(2) MWI and CI can be viewed as approximations to decoherence. Decoherence has a time-scale on which it occurs. This time-scale is different from anything in classical systems. The dead and live copies of Schrodinger's cat can in principle interfere with each other, even after days have passed -- the interference effects just fall off exponentially, with such a short time-scale, that it becomes impractical to observe them.
