Achilles for all his prowess in a fight, is as far as I can remember, wasn't a demigod in Troy. He was just insanely skilled. Possibly it was alluded to but not outright stated like how Perseus is Zeus' son and stated to be a demi-god multiple times. Feel free to correct me if I'm wrong, but even if he was, he only ever fought humans.
Perseus on the other hand is a Demi-God who is powerful enough to fight full-fledged Gods like Ares, as well as mythical creatures like the Scorpioch, a Chimera and the Minotaur, as well as superhumans like Calibos. By feats he's stronger, tougher, and faster than Achilles even if he's not quite as skilled. His stats give him a huge edge.
And if Achilles gets some sort of God blessed armor(which I can't remember if he actually had or not), then Perseus should get his god gear like his Lightning Sword and the Scorpion Shield.