What are some differences of nuclear and chemical rocket engines?

The major weakness of a nuclear rocket is that it is nuclear. Today there are folks who get upset at the launching of spacecraft with RTGs. (And not totally without reason. See that red glowing pellet on the Wiki page? Would you like that crashing through your house? What if someone recovered it and used it a “dirty bomb”? Etc..) Frankly, a vehicle with a simple nuclear rocket as a first stage simply would not be politically viable at all.

A simple nuclear rocket discharges fission daughter products into the exhausted propellant which is heated by passing through the reactor core. This is indeed a rather nasty prospect if you intend to use it with the Earth’s atmosphere. As you can heat the propellant much hotter this way, the exhaust velocity is increased in comparison to chemical rockets. But that nuclear core is heavy. If you change to a design that avoids direct contact between the core and the propellant then things get heavier still and much more complicated.