Well, it's a lot of things. For one, she has sustain for absolute centuries. As long as she can land her skill shots, she's going to just heal herself back up, so that's one problem people have with her kit.
Next, is her ability to rip your soul out. No matter what you do, half the time you can't stop her from slowing you and making you a vessel, so if you do survive it, you have to kill tentacles while they smack you in the mouth, that or wait out the full SIXTY SECOND DURATION, or Kill Illaoi.
Most people also underestimate the range of her tentacles when she uses harsh lesson; some people don't really realize just how far from the wall they have to be, and it can lead to some heavy chunk damage if you're not careful.
Her ult also deals way too much damage in team fights, at least for how tanky Illaoi gets. Basically a lot of her mechanics can't really be countered. What, are you just supposed to not group up for team fights and let them take your shit for free? Are you supposed to 3v5 just because if you don't her ult is going to destroy you if there are 5 enemy champions in the same general area as Illaoi? I mean, you can try to stand back, but its got a huge range. Not to mention it makes her damn near unkillable because all of the untargetable tentacles will still heal her for a percent of her missing health anyway. Also, when your soul is ripped out, what do you do??? Enage? Well, that's stupid because it does percent health damage whether she kills it or not, and is going to out trade you, if not outright kill you. If she falls behind she doesn't do very well, but usually she stays ahead if you don't camp her.