People who die by drowning as a general rule don't die because their lungs fill with water. Actually, very little water, either salt or fresh, enters the lungs before death. Instead, drowning victims usually experience laryngospasm, the body's defense against water entering the lungs.
Laryngospasm is when a person's vocal cords spasm and close off their larynx or windpipe, blocking the flow of air or, in this case, water. It doesn't matter whether the water is fresh or salty; the crucial point is, when liquid of any sort enters the larynx the body reflexively closes it off.
Any water found in a drowning victim's lungs generally seeps in after death, when all the muscles in the body relax. Therefore, the osmolarity of the liquid should not make a difference in the amount of pain experienced while dying.
I would tend to consider it an urban legend that it is less painful to drown in salt water, since anyone who could speak as an authority on the subject would have to have been drowned in first one and then the other liquid.
The experience related by C-Dawg is called infiltration. It happens when an IV catheter slips out of the vein and the IV solution flows into the surrounding tissue instead. The pain is caused not by the salinity of the solution but by the increased pressure being locally experienced in the tissue. Our bodies are not simply bags of fluid, but an array of small, individual, mostly inelastic compartments, which generally stay at a fixed pressure. A sudden increase in pressure will always cause pain.