There's a massive difference between the two. The Welsh, Scottish and Northern Irish (plus islanders and whatnot) tend to get pretty peeved when you refer to them as English.
I accept that people outside the UK might not care, but I don't think that's necessarily the case. There's also a significant difference between making the mistake in casual conversation and on a formal document.
There's not a massive difference. These are people who speak (for the most part) the same language, have (for the most part) the same religion (Christianity), live a few miles from each others, listen to the same music, watch the same tv shows and read similar books at home and school. Sorry, but for outsiders, you are the same people and for the most part, they'd be right to think so. People are not that different, especially people who have so much in common as the inhabitants of the UK, get over yourself, you're not that special.
I'm sure you don't know and care that southern Peruvians from Arequipa like to be called Arequipaños and wouldn't like being mistaken with Northern Peruvian from the Jungle etc. Thing is they're the same to any outsider.
Calling all Americans Yankees is exactly like calling all Britons English. It's not a slur unless used as one (Yankees are, roughly, people from New England/northeast U.S.) but many people will take offense when incorrectly labelled so - even if "no one cares". You could imagine a Breton's reaction to being called a Parisian to be roughly similar. Or a self-professed Catalonian being described as Castilian. Or an American asking about best place to get currywurst or bangers and mash in Paris, I mean all these places are pretty similar anyway, y'all listen to Eurovision, sorry if that offends you...
I accept that people outside the UK might not care, but I don't think that's necessarily the case. There's also a significant difference between making the mistake in casual conversation and on a formal document.