-Illaoi has no hard CC (her only CC is in the form of Damage or a sometimes situational slow on E which is 80% for 2 secs)
-Her engage is minimal (through a spirit pull, flash, or simply walking in)
-She has no disengage, but serves as an anti-engage (as in, she can't easily leave the fight, but the fight may leave her).
-Her mobility to limited to a short ranged dash (which CAN be used over walls in some situations)
-Illaoi sustains herself through her tentacles (5% missing HP per tentacle, generally every 4-2secs)
-Illaoi's passive supports a "pick a spot" or "presence creep" playstyle where she claims a spot and keeps you out while also trying to encroach into your area.
-Illaoi's ult creates a huge "NO-TRESPASSING" area, which scales per enemy champion (and her spirit). It will deal massive damage if enemies remain in the area (while sustaining herself as a tank), however, this area does NOT move so if she starts a fight or anti-engages with it, her increased damage is more limited to that zone. This means that once the ultimate goes off, it is highly recommended to leave the area if possible. otherwise she should be the focus on that fight (since you want to focus the highest damage dealing threat possible anyway).
-Darius also has limited CC but still more easily applied than Illaoi. His crippling strike has a slow (90% for 1 sec). He has a medium ranged pull which is followed by another 40% slow for 1sec.
-Darius engage is minimal, but more consistant or reliable than Illaoi due to his W and E slows/pull being easier to apply than the Spirit Pull.
-Darius has minimal disengage due to his W and E having easy to apply slows, while his Q also doesnt restrict his movements and can potentially heal him while disengaging.
-Darius has sustain on his Q, but is less than Illaoi's in a team fight due to the restriction and CD (it maxes at 36% missing HP and has a CD ranging from 9sec to 3sec. Illaoi's is maxed by number of tentacle hits.)
-Darius passive lets him all-in on champions he can pick a fight with through his W or E, while also giving him the ability to hit a few times and back off to slowly let them bleed down for that all-in or to force them out of the area for fear of it.
-Darius ult can be used to pick off weak opponents that he or another teammate damaged, gets a CD refund and can be reused on another champion. Combined with his passive (Noxian Might), his ult can deal about 500 true damage instantly late game. His ult however, doesnt restrict his ability to engage or disengage.
-While bursting her down is possible through raw damage and LORD DOMINIKS, having anti-sustain items like MORELLO, EXECUTIONER or MORTAL REMINDER are REQUIRED if you are unable to burst her down completely due to her being naturally tanky or building additional tank items (watch for the steraks against burst comps).
-Poking her before a teamfight may increase your chances of a successful (re)engage and as mentioned above grievous wounds is extremely valuable. This means ADCs like: Ezreal, Caitlyn, Twitch, Varus, Ashe and Tristana are good picks when played well and when you using your range properly [EDIT: Should note that Death's Dance helps counter poke if minions are around]. For more teamfight base ADCs, i would suggest kogmaw (or Ashe again), Lucian or miss fortune.
-Mid champs like Ziggs, Syndra, Velkoz, Orianna, and Veigar, while squishy, have strong CC and/or strong poke for either teamfights or for directly dealing with her. Casseopia is also a strong pick due to her high DPS and slow with Rylais, while her MIASMA will prevent Illaoi's W's from being used which makes the tentacles slam (Miasma lasts for 5 secs, Illaoi's ult lasts for 8. think about that.)
-Top champs like Vlad, Olaf, Tryn, Swain, Ryze are strong 1v1 champs who can poke and/or all-in very well on illaoi early or late, but i would suggesting taking IGNITE to help guarantee any potential kills solo or with jg's help. Or potentially build an EXECUTIONER/MORELLO as a 1st or 2nd item. The typical Juggernauts (Darius/Garen/Mundo/Nasus) can work very well also when played well.
End game damage was on par with our Kog'maw, both ~15k higher than the next.
Keep in mind that Illaoi's damage is nearly completely AOE based. So while 10k damage looks nice, its spread throughout 5 enemies, which may not even die for various reasons. Where as a kogmaw focusing down priority targets dealing 10k damage, will more than likely guarantee a couple kills.
This is basically a tank doing Kog'maw-level damage. How the fuck is this okay?
Illaoi is a juggernaut. She is a heavy duty, high damage, team fighter who can withstand high damage with sustain. a Tank is a champion with hard and soft CC who deals low-moderate damage but is able to withstand high damage without sustain but can also peel/engage/disengage for the team. some tanks also have utility.
Decided to take a look at the game you mentioned and the only form of grievous wounds were two ignites between Darius and Azir. No Lord Dominiks or Mortal Reminder were on your team. Yes Kogmaw ignores 28% resists, but you had no increased burst from Dominiks, or anti-sustain with Executioner/Mortal. And you can't always guarantee your team would always have ignite up only for Illaoi, especially with other potential priority targets and the chance of both ignites being used at once in the first fight, which leaves no counters in the second. You would of definitely got more value from the Executioner than the Jaurims fist, the extra health/slow from Frozen Mallet would of been nice, but Illaoi alone would just outheal most of the damage by attacking other targets around her (if she couldnt get to you) nevertheless Azir had Rylai's for his Soldiers to proc which meant you had a slot to sacrifice.
Like, i'm sorry you had a rough game against her, you (or Darius/Ezreal) should of planned/thought ahead to get Executioner/Giant slayer as soon as you could without sacrificing too much especially since you would apply it more often than Ezreal could. Rageblade could of been delayed, Jaurim's first could of been either Executioner or Giant Slayer. Mistakes were made and Illaoi tested you. Hope you learn how to play around her soon!