Armed with swords, Conan could force a tie. Barely.
Bare-handed, Tarzan has three options: Let Conan live (unconscious), force Conan to submit, or kill Conan. Conan just can't get Tarzan into a position where Conan's marginally superior strength can be the decisive factor - Tarzan is nimbler, more savage, and has decades of experience killing foes that are stronger, but not so wily as Tarzan. Conan had better hope for mercy from Tarzan.
Another way to put the difference: If Predators landed and hunted Conan, Conan would win, after convincing his chosen foe he was dead and then pullling off a deadly sneak attack.. and the remaining Predator team would appear, salute, enter their ship, and they'd take off. If the Predators tried that with Tarzan, both surviving Predators would flee Earth, leaking fluids - and then discover Tarzan was on their ship with them.
Yet another way: Conan would be an old school, hardcore Fighter, with plenty of Barbarian. Yet he faces a nightmare: an old school Grandmaster of Flowers dual-classed to Barbarian, and full of old-school buttkicking goodness. Tarzan has killed apes, tigers, lions, giant snakes, Nazis, duellists, and dinosaurs with his bare hands before. Conan has killed his share, but not for as many years (immortality does matter).
Draka wish they could grow up to be like Tarzan.