I've always thought that the primary reason for that was so that they could avoid surfacing for months (or even years?) at a time to remain truly untrackable. The US and UK use nuclear submarines as part of their nuclear deterrent--they carry nuclear missiles and could launch a revenge strike against Russia (or any other hypothetical country... but really Russia) even if they somehow "won" the ICBM exchange and just obliterated everything in the US and UK, including missiles in silos and bombs waiting to loaded onto bombers.

A diesel sub can't do that--they can stay underwater for at most a few weeks at a time. A nuclear sub can leave its base and just loiter literally anywhere in the ocean. It's not about being sneakier. It's about being anywhere.