What advantage does nuclear power have over fossil fuels?

In practice, remember that all fuel sources of energy are thermal, and the conversion to electrical power uses the same mechanisms, and therefore similar efficiencies. We do not use “nuclear power” any more than we do “coal power”.

BOTH generate thermal energy – although by different processes – that is used to generate steam, which is used to generate electricity through expansion turbines and generators. Gas turbines, combined-cycle and internal combustion directly drive generators without steam production.

Every power generation technology has its own unique advantages and disadvantages. There are reasons that we use all of them. Improvements can also be made in the overall efficiency (lifecycle) of all of them, and we should be very reluctant to discard any of them entirely, or to rely on any one of them excessively.
You can see from the references that different definitions of “efficiency” may be used promote a particular outcome.
http://www.nei.org/Master-Document-Folder/Backgrounders/Fact-Sheets/Quick-Facts-Nuclear-Energy-in-America

Nuclear energy facilities generate electricity 24/7 at a 92 percent capacity factor. This is more efficient than other types of energy—combined-cycle natural gas, with a 47.8 percent capacity factor; coal-fired at 60.9 percent; and wind at 33.9 percent.
https://www.eia.gov/tools/faqs/faq.cfm?id=107&t=3
and heat rates (efficiencies) here:
http://www.eia.gov/electricity/annual/html/epa_08_02.html