What explanatory power (or even interpretational satisfaction) do models such as many worlds give us that the laws of quantum mechanics with decoherence alone do not?
It should be noted that interpretation of quantum mechanics do not provide any explanatory power. An interpretation is a way for us to try to "intuitively" understand what the equations are telling us, and as such might help to guide further research, but in itself, it does not explain anything.
In this sense "decoherence", in the sense of the work of people such as for example Zurek et al., might be a little different than many-world, in that it attempts to model the mechanism for which a pure state becomes a "classical" mixture. In this sense, it is less an interpretation than a model for the measurement process.
Still, I would argue that decoherence models do not really "explain" the collapse of the wavefunction, as the interpretation of the eigenvalues of the density matrix as outcome probabilities still relies on the Born rule in the larger system+environment space. It might explain why, neglecting the larger environment, measurement results have a well-defined value, but it does not explain how one the possible outcomes is "chosen" any more than Born's rule/axiom does.
More generally, I don't think there is any commonly agreed-upon answer to the question "are there still gaps in our knowledge of quantum mechanics that need further explanation". Many would argue that the answer is yes, others that it is no. We will likely only know that the answer was yes if/when someone comes up with a better theory.