A saltwater pool and a chlorine pool have almost all of the same components. The main structure -- the basin -- is identical in both pools. The difference is that while chlorine pools have a system for filtering and chlorinating the water, saltwater pools have chlorination units that break down salt into chlorine via electrolysis and adds the chlorine to the water. Thus, the basic pool structure will cost you the same, depending on what material you choose to line the pool with and what size you choose for your pool.
Expect to pay more for the mechanism that runs a salt pool -- the chlorination unit -- than for the filtering mechanism in a chlorine pool. That's because while a chlorine pool's filtering unit is only filtering the chlorine you add, the chlorination unit in a saltwater pool is essentially like a mini chlorine factory, turning salt into chlorine and then filtering the chlorine before releasing it into the pool water. Expect to spend a few thousand dollars extra for the mechanism necessary to turn salt into chlorine.
Chlorine pools often have many parts made of metal. Salt, however, can corrode metal, so saltwater pools need components such as pumps and pipes that resist corrosion. If you're building a pool from the ground up, this shouldn't add to the cost because alternate materials, like plastics, aren't more expensive than metal components. Replacing metal components for noncorrosive ones can be a costly endeavor.
Saltwater pools are generally considered less expensive to run, even from the start, because the salt needed to operate the pool is significantly less expensive than the chlorine added to chlorine pools. In addition, salt never disappears from the water once it's added, so the amount that needs to be added after set up is a lot less than the amount that is initially added.