Why do mining pools require 120 confirms for a solved block before payout? According to this question here, 6 confirmations can be considered 'secure enough'. Some pools out there also offer payout at 100 confirms but that's still substantially higher than 6.

Why is such a high confirmation count required for this?
There is a protocol rule which prohibits generated BTC from being spent until it has 100 confirmations. The Satoshi client has an additional safety margin: it won't let you spend generated BTC until it has 120 confirmations. The purpose of this rule is to prevent long chain splits from invalidating lots of transactions that spend newly-generated BTC. For example, this rule prevented non-coinbase transactions from being invalidated after the overflow incident invalidated 50+ blocks.

Some pools (e.g. Eligius) pay people by putting the payments in generation transactions. The recipients still won't be able to spend this BTC for 100 blocks, but at least the payments will appear in their Bitcoin clients right away.