Perhaps the most significant benefit of the approach we took was that it enabled us to converge GitHub for Mac and GitHub for Windows into GitHub Desktop iteratively, rather than as a ground-up rewrite.
Enabling iteration can really improve the momentum and also the end result of a refactoring. Experiences with the new ways can be had early and the project itself can live and evolve in parallel with the refactoring rather than stall until the entire app has been rebuilt in the new way. Enabling iterative development is something I feel improves most projects. The validation and reward of seeing your changes live and working early has a good impact on both code and developers.