Advantages include an emmission-free power source (not counting the emmissions from mining and refining uranium and constructing the plant) and reliable energy compared to other emmission-free sources (wind, solar). It won't help with energy independence, since we get almost none of our electricity from burning oil.

Disadvantages: non-renewable (depends on uranium, which is finite just like oil), generates waste that is dangerous for 100s of years (and they're gonna build the waste dump in YOUR back yard), a nuke plant is a huge terrorist target (so are radioactive waste dumps and uranium processing facilities), nuke plants take YEARS if not DECADES to permit and build. And oh yeah, it's HUGELY expensive.