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Who wins?

Both with regular swords of their preference.

Just swords?

Achilles stomps.

Achilles armor is blessed by the Gods. A regular sword will not cut it. Pun intended.

Give Perseus the sword given to him by his father, and Perseus obliterates Achilles.

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.

This is a horrible mismatch. Perseus seriously outstats Achilles in all but skill

Perseus was fighting people that landed from thousands of feet up.