That’s not true anymore, since the switch to AWS you get raw IRC via irc.chat.twitch.tv (or irc.twitch.tv) and IRC over Websocket via irc-ws.chat.twitch.tv. The ports don’t matter for that distinction anymore.
On the original topic: I can connect to irc.chat.twitch.tv using raw IRC (no Websockets) just fine, on all 4 ports: 80, 6667, 443 (SSL), 6697 (SSL)
Both irc.chat.twitch.tv and irc.twitch.tv resolve to the same IPs, so there shouldn’t be much difference. It’s possible that the IPs are in a different order so that you get a different IP if you just request one from your environment and that that specific server has issues. I’ve also heard of an issue with the certificate when connecting to irc.twitch.tv via SSL.
Either way, if these issues persist it would be useful to know which IP exactly you are trying to connect to, so others can test if they are running into the same problem.