Alistar's main problem IMO is that, while he is among the gods at purely soaking up damage and not dying, he lacks compared to other tanks around him at initiating. Every tank (and even "offtanks) commonly ranked above him all have some kind of initiation that doesn't require physically trying to run in from afar. He definitely has a much harder time dealing with kiting than Mumu, Shen, Rammus, etc and this greatly hurts him.
Now, past that you have dealing with his laning phase. A lot of tanks above him can simple forgo regular laning and jungle (again, mumu shen Ram) and multi task for the team where as Alistar will be sent babysitting whoever in the duo lane. Then you have general utility for the team that he lacks. Shen Global Ult/taunt saves, Mumu AoE ****, Rammus super fast powerball mode/free migitation, etc. Then you have the fact that other tanks can do a surprising amount of damage with little effort. Mumu obv with his AoE Health % drain, Ram with his ult and extra damage taken from taunt, etc. Alistar can disable people and soak up damage like a sponge, but he's also not real threatening damage wise. Something bad like Tank Mundo probably does a LOT more damage than Alistar would be expected to do. Because he doesn't do that much damage, after he blows his combo he's frankly pretty ignorable for the rest of the fight til his CD's come back up. With other tanks, even after Mumu does Bandage to Ult, you STILL have to worry about him running around melting your HP in an AoE. Or after Rammus dives in with powerball and taunt, you still have his ult to worry about. Alistar doesn't have anything like that after his combo, which is a shame.
Alistar is just outshined in a lot of areas for the tank role. His ability to soak up damage probably better than anyone else in the game is definitely overshadowed by his flaws. That's my take on him anyways.