"It appears to me that branching is equivalent to collapse... so trying to see what I'm misunderstanding."
The idea of branching universes in the popular picture is indeed equivalent to collapse. However, the Everett Interpretation does not have branching universes; it says that when a quantum observer interacts with a system they jointly evolve into a superposition of orthogonal (and hence mutually non-interacting) states, in each of which the observer sees one outcome. From the point of view of the quantum observer, it is as if they were in separate worlds.
Back in the 1970s when Bryce DeWitt was trying to raise awareness of Everett's idea with the public, he used this 'Many Worlds' image to explain how it would appear to someone living in such a universe. But it's not how the physics works, and taking it too literally causes endless confusion.
"So we start with the wave function... Alice measures the spin on her particle. It can be spin up or spin down. By many worlds, the wave function branches into two at this measurement?"
You start with two particles which are jointly in a superposition, so their up/down spins are correlated. When Alice interacts with her particle, her wavefunction becomes correlated with that of the particle, and Alice herself becomes a superposition of an Alice seeing $\left|\uparrow \right>$ and an Alice seeing $\left|\downarrow \right>$. Nothing happens to the other particle, no changes travel outside the locality - whether faster-than-light or otherwise.
Using the 'Many Worlds' analogy, when the particles are prepared in a superposition of up/down states, this is when the particles first split across two worlds. When Alice measures her particle, she splits to join it, one version of her moving to one of the particles' worlds, the other version of her moving into the other.
Either before or after Alice makes her measurement, it doesn't matter which, Bob measures the partner particle, and he too enters a superposition of a Bob seeing $\left|\downarrow \right>$ and a Bob seeing $\left|\uparrow \right>$.
Bob's state is correlated to his particle, which is correlated to Alice's particle, which is correlated to Alice. So Bob and Alice are now correlated in exactly the same sort of way as the original particles were correlated.
And so when Alice and Bob get back together to compare notes, the version of Alice that saw $\left|\uparrow \right>$ can only see/interact with the version of Bob that saw $\left|\downarrow \right>$, and vice versa. They have both split and joined the two particles in their two 'worlds', and each only sees the Alice or Bob in the same 'world' as themselves.
"I can't really see the difference... Unless the branches existed prior to measurements happening... then Alice is just finding out which branch she was always on."
The 'branches' existed prior to the measurements happening, but Alice only split and entered the two branches when the measurement was made. She was not 'on' either branch until that point. The idea of branching universes only really makes sense when there is only a single observer in the universe. As soon as you try to deal with multiple observers, the analogy falls apart. The split is observer-specific. One observer may be split without affecting any other observers.
"So what exactly is the advantage of postulating the existence of these other branches?"
Philosophical elegance and simplicity. Superpositions already exist and are required in all consistent-with-experiment theories of quantum mechanics - all we are doing is extending the same theory unchanged to the macroscopic level. It's the same theory but without collapse, so it's simpler. It's also local, deterministic, reversible, conserves information, and realist. We don't need to explain how collapse works or what triggers it. We don't need to deal with having separate 'quantum' and 'classical' parts of the universe, operating according to different rules, and a mysterious and badly-defined boundary between them. There is the potential to explain features of quantum mechanics in terms of underlying mechanisms that would otherwise simply have to be asserted as axioms, and removes the need to explain other features (like non-determinism) entirely. It's more explanatory.
"What I mean is... from an empirical point of view... we observe a measurement... now we could have split into several branches... but we can never observe this for sure."
Correct. The 'unitary evolution' part of quantum mechanics predicts a superposition of states, almost all of which we do not and cannot see. Copenhagen says all but one of them disappears at some unspecified time during a measurement, by some unexplained mechanism. There is no way (if quantum mechanics is correct) to distinguish between them experimentally. The Scientific Method is entirely agnostic on the point.
It all comes down to how you want to wield Occam's Razor. Do you want to remove the gazillions of unobservable states as unnecessary, or the unexplained and unobservable mechanisms by which 'wavefunction collapse' invisibly deletes all the unseen entities that the simpler theory predicts, that you could never have seen anyway? Neither is 'necessary', given the alternative of the other.
