Nuclear powered propulsion systems are needed if we are to become an interplanetary civilization - if we are to cross the large distances between planets effectively.
Chemical power may have been enough for car and planes, but it's far from sufficient if we are to ever develop some form of interplanetary commerce. It just doesn't generate enough energy.
Today's chemical powered rockets are very expansive and they can barely reach the orbit. A chemical rocket will transport only a few astronauts to Mars in 6 months, after billions are spent on a single mission.
The chemical propulsion's potential is exhausted.
You want to colonize Mars, to move millions of people there? You want to exploit the asteroid belt? Even with a space elevator, you couldn't do that in less than centuries with chemically propelled interplanetary craft.