I've been maining her in my climb through silver and gold this season and in norms right after she came out. I'll agree with what a lot of people are saying her and she definitely has a spot in counter-engage and anti-dive comps. Even when you're ahead turret dives on Illaoi can turn into ritual suicide incredibly quickly. Also kind if in line with Galio and Annie her Ult-Flash combo is incredibly powerful if you see the opportunity and can be used to initiate in certain situations, it kind of brings up the "scatter or die" response from the enemy team which can be great for getting a couple of picks or flashes and then brute forcing an objective with a numbers or cooldowns advantage. She's also good at running interference for a Drag or Baron play by zoning out choke points with tentacles and a threatening ult. Even if you ult and they back away that buys at least 8 seconds for your team to complete the objective and if they don't back off well you just ulted in the middle of their team. As far as individual lane matchups go Tryndamere and Nasus are your worst nightmares. Nasus because of his passive lifeteal and Q stacking off your tentacles makes him very hard to effectively push out of lane, he practically wants you to land E on him for more lasthits. After a certain point of the lane he's going to be too tanky in relation to you and the free Q stacks he picked up from your harass are going to kick and you're going to take a lot of damage, not to mention when you ult he can just wither you and walk away. Tryndamere just gets a free 6 seconds of life no matter how hard you burst him with your ult and if worst comes to worst for him he can just spin out, but usually if you trade ults you die and he lives anyways. Obviously skill is the biggest factor and if you can outplay whomever you're laning against then even if they're on Trynd or Nasus you're probably going to win lane, but just try to avoid Trynd and Nasus. A lot of people say that ranged champs like Quinn and Lulu hardcounter her but I have not had exceptional difficulty with either of them. Basically just play passive early, but once you hit 6 you can kind of turn the bullying around because they're going to take a lot more damage than you are and if they go for the kill your ulted W actually gives really solid chasing potential. The E is also incredibly important in ranged matchups, if you can land one you don't even have to kill it but you can three free tentacle hits and a grasp of the undying proc on it to very easily sustain through the lane. Most melee champions are a complete faceroll at least in gold/silver for a general lack of understanding of illaoi mechanics and other than that she just has a kit designed to bully. I've actually had like a 67% win rate climbing to gold 2 from silver 4 with her this season and that's with her as more than half of my total ranked games.
tl;dr She's decent against most comps aside from heavy kiting/poke, avoid laning against Nasus and Tryndamere.