As any collapse (if it does happen) occurs so ‘late’ that current experiments are unable to differentiate between many worlds and collapse—it seems quite possible that both theories will continue to give identical predictions for all realisable situations, with the only difference being ‘one branch becomes realised’ and ‘all branches become realised’.
Assuming this practical indistinguishability between the theories, I think that any utility function based on one of the theories can be directly translated into the other theory by just reinterpreting the theory-inherent probabilities. This assumes that all branches in the many worlds reasoning are weighted with their ‘probability’ (e.g. the Quantum Russian Roulette thought experiment hinges on counting ‘I survive’-branches differently¹)
More Human related:
One relevant aspect is how natural utility maximisation feels using one of the two theories as world model. Thinking in many worlds terms makes expected utility maximisation a lot more vivid compared to the different future outcomes being ‘mere probabilities’—on the other hand, this vividness makes rationalisation of pre-existing intuitions easier.
Another point is that most people strongly value existence/non-existence additionally to the quality and ‘probability’ of existence (e.g. people might play Quantum Russian Roulette but not normal Russian Roulette as many worlds makes sure that they will survive [in some branches]). This makes many worlds feel more comforting when facing high probabilities of grim futures.
A third aspect is the consequences for the concept of identity. Adopting many worlds as world model also means that naive models of self and identity are up for a major revision. As argued above, valuing all future branch selves equally (=weighted by the ‘probabilities’) should make many worlds and collapse equivalent (up to the ‘certain survival [in some branches]’ aspect). A different choice in accounting for many worlds might not be translatable into the collapse world model.
I am still very much confused by decision theories that involve coordination without a causal link between agents such as Multiverse-wide Cooperation. For such theories, other considerations might also be important.
¹: To be more exact, I would argue that the case for Quantum Russian Roulette becomes identical to the case for normal Russian Roulette if many world branches are weighted with their ‘probabilities’ and also takes into account the ‘certain survival [in some branches]’ bonus that many worlds gives.