Why is a nuclear submarine better than a diesel submarine?

Though Diesel-Electric boats are in fact much quieter on the battery than Nuclear Powered boats (the first thing taught in Sonar school), even with today's battery technology, DE submarines are limited in range, equipment, speed, maneuverability, and weapons they can carry. This in effect limits their overall usage as a weapons/intelligence/covert insertion/Naval escort platform.

In the early '60's, there was a big debate over the cost of nuclear powered boats vs. conventional DE submarines, as the success of the USS Nautilus (SSN-571) led the Navy to push for more of its type. A tactical test pitted one NP boat against several DE submarines - in simulated attacks, all DE boats were discovered and sunk by the NP submarine. This is what led to the Navy's current doctrine of using only nuclear powered boats.

One of the largest drawbacks to DE technology today is the major advancements in Anti-Submarine Warfare (ASW) technology, which has developed considerably over the years since WWII. Though the biggest threat to any submarine is still a submarine, patrol aircraft (fixed-wing and rotor) use sophisticated sonobuoys and dipping sonars that can direct surface and other air assets to the area to track and destroy a submarine. In such cases, the ability to go deep and rapidly egress the area is paramount, and the speeds/depths of modern NP submarines allows them to do that, where DE submarine technology is limited in that regard.