Yu Shu Lien wields a dao (which is a saber that can be split into two smaller blades), a spear, dual hook swords, a metal club, and a broad sword. She does so in that order. Of note, Yu Shu Lien also grabs a monk's spade at one point but it turns out to be too heavy for her to manage correctly.
Jen Yu just uses the Green Destiny, which is a kind of magical blade that can just cut through anything (including stone) and never breaks. So it's a cheat weapon.
In the scene Jen is angry at everyone in her life who's been pressuring her, and she's also super eager to prove herself worthy of wielding the sword. Shu Lien has just told Jen that they have hidden Lo, Jen's bandit ex-boyfriend, in the Wudan Mountains, and Jen is extremely flippant at this news. Shu Lien has no f*cks left to give and is tired of Jen's attitude so a fight breaks out, and it's amazing.
Shu Lien is a way better fighter using inferior tools, and so they're basically equal. As a veteran warrior, it's mesmerizing to watch Michelle Yeoh's character grab weapon after weapon to use against the sword that just slices through them like they are nothing. If she was going to kill Jen, she could have beat her. But that's not what's going on. She says as much too with the line, "Without the Green Destiny, you are nothing." Being a Brat, Jen fails to accept defeat gracefully and ends up hurting Shu Lien in a minor way before fleeing.