So let's clear this up shall we.
First up what is ilvl anyhow?
ilvl, or item level, is simply the average of the level of all of your gear. Each piece of gear, or item, has a power level. Each player has a potential of 16 items (2-handed weapons count twice unless its something like titan grip...but anyhow) so we add up all the items levels and divide by 16 to get the players equipped item level. Your character sheet and armory entry will also have the average of the highest level pieces you own and could equip if you so desired. (as an aside, this fact used to be used to cheat ilvl restrictions on dungeons by loaning people boe gear to get them into the dungeon and having them return said gear later).
Got it? Good but this doesn't tell us what ilvl is just how its computed.
So each item has a power number, that is its item, which determines the budget used for assigning stats to the item. This is one of the ways Blizzard balances things and is also why when they screw up you get people using trinkets that are 10, 20, 100 ilvls off.
So on the face of it a player with an ilvl of 655 should be better than one with an ilvl of 630. They have more stats allocated to them and since stats are the variables used to fill in our damage and healing numbers bigger ilvl must mean better performance.
You see ilvl assumes two things and it turns out when those two things are false the entire ilvl system falls apart.
A players gear will be itemized appropriately for the roll, spec, and class of the player.
Players know how to play their characters properly in order to best utilize their itemized stats.
Once either one of those assumptions turns out to be false you loose any meaning to ilvl. This is why ranking aren't simply a top down list of item level.
Looking at Feral Druids ranked by WCL for Mythic Butcher on the date this was posted we see this:
The top ranked Druid did 40.4k DPS at ilvl 679
The top Druid of the top ilvl's did 37.8k at 684
The top of the bottom was 34.8k at 672
so their dmg per ilvl was : 59.5; 55.3; and 51.8.
What accounts for this? Well, without looking at armory pages or fight logs we can guess that some players had gear that was better itemized for their spec and some simply played better.
Now it is true that there is a minimum ilvl needed in order to complete these encounters. If a group wearing starter gear can beat Mythic Kargath then I will concede that raiding is too easy, challenge extended.
The fact is that most of us have poorly itemized gear which means that we have a lower potential performance per item level number to begin with and further we don't play perfectly ....ever. There may be a few people out there who can every possible bit of damage out of their gear but those people are rare. Even the guys in Paragon or Method or Giggity for that matter are only getting about 90% out of their gear. Average joes like you and I....we are more likely lucky to get 50% out of our gear.
But I digress.
The problem is even worse for non-DPS specs. If a tank can handle an encounter at 630 then he's basically done. He won, checkmate, no gear needed. Adding ilvls to tanks works to reduce strain on healers, speed up the kill meaning less stress on DPS, and allows for more stacks to be taken meaning less time spent on the edge of perfection.
For healers it means each point of mana is converted into more healing. If you are overhealing at 630 then you aren't using your mana wisely to begin with, leveling up to 680 just means your overhealing will be that much larger. Furthermore your ability to top charts is directly reliant on other peoples ability to make mistakes. The amount of avoidable damage a raid takes is set, predetermined, sure it'll be a bit less the faster you beat the boss but if you can heal all the avoidable dmg in ilvl 630 gear then 680 aint doin you a bit of good. You need people to stand in the stupid, tanks to flub their mitigation, mechanics to be borked in order to inflate your numbers and justify that tasty, tasty loot.
DPS get the straightest conversion out of all roles given that assuming all other factors being equal if ilvl goes up then the amount of dmg you deal also goes up. The boss dies faster so that bigger damage number is divided into fewer time numbers and you get nicer DPS numbers to gloat about. The better you execute your rotation, the less you are in danger of dying or moving, the more you can get out of your gear. Failure to use pots or build up enough blah blahs or even to live means your DPS number begins to sink. Suddenly you are looking at people 15 ilvls lower who are out DPSing you and wondering why you suck so bad.
So ilvl is relative number that means very little at the end of the day. Until we all perform perfectly in perfectly itemized gear ilvl really doesn't mean much of anything. If you are worried that 620 is too low to join your guild in Normal Highmaul then learn to get 630 worth of healing from that 620 gear. Out play your ilvl and no one will ever know that you were undergeared.