Why is the Trakt schedule differ from the actual air date?

For example Trakt shows 'Gilded Age episode 2 airs at 11:00pm on 1/30. But HBO shows it airs on 1/31?

Trakt converts all airing time information to your own timezone, so that might be the “problem”.

Could be, if it was only an hour or so difference, but 24 hours?

Yes, please check your timezone is set correctly at https://trakt.tv/settings and that your device timezone is also correct. That’s what the issue like to me as well.