As I said, he used a parachute to survive the crash with the helecopter.
And Snake was much more damaged, tired and wounded when they fought. And Snake had just previousley fainted. Have you ever fainted? You feel sick and have a hard time standing up, still he kicks Liquid's ass. Yeah, they were pretty equal, but Snake proved faster and stronger in the end. Did you see that punch? It made him fly mabey 4 meters up in the air and far away.... It was the same terrain for both, Snake had same chance of falling down as Liquid, but Snake overcame him...