I don't want to fall out with anyone over this, but Connacht is the name in English as well. Connaught is the name of a hotel, and indeed one of the local newspapers keeps this nostalgic spelling 'The Connaught Tribune' (the other is the Connacht Telegraph), but no literate person would seriously think of spelling the province in any way other than Connacht nowadays.
But as I say, it's not something to fall out about. If you prefer the Anglo spelling harking back to the days of Empire, stick with it and we can still be friends.