Everything in the U.S. fleet except the large aircraft carriers and the submarines burns crappy low-grade diesel fuel. For the submarines, there are major advantages to not having oxygen-requiring engines, and for the aircraft carriers, they want to carry a lot of fuel for aircraft, not themselves. On the other hand nuclear reactors are too expensive to merit putting in ships which don't require them.

That is part of it. The other part is that for smaller ships, you end up having over half the ship manned by sailors who can handle the engine room and reactor. The Navy has enough trouble finding and keeping sailors for their nuclear power program, so they tend to restrict how they use them, otherwise they might end up in the unfortunate situation of needing to allow less qualified candidates through nuclear power school just to keep the fleet running.