Mix and match depending what suits your child. We've done abrsm and trinity here at different times. Check your syllabus though on line- don't just trust the teacher if you are mixing boards- dd1 had to go back and sing a folk song off the top of her head as they'd missed that out ! It was fine.
Dd2 found trinity theory much easier. Just suited her style.
Strangely trinity seem to do "grade books" for higher grades than abrsm do, which if you have bit of a budget to stick to can make a huge difference compared to buying a book or each individual piece.
Trinity and ABRSM are the big ones for classical music. I'm not sure whether they do other things as well. They are equally difficult and respected, in my experience at least. Rockschool ones are validated by one of those two, or at least they were a few years ago when DS did one. All of them can potentially count for UCAS points, though it's best not to rely on that as not all universities use that tariff system.
There's nothing to stop you switching between them, but the sylabuses won't be quite the same.
The ones I know of and have used are ABRSM, which is the largest and probably most well known, Trinity Guildhall, London College of Music and Rockschool.
TG, in addition to classical also does rock exams (new syllabus).
ABRSM has jazz exams up to grade 5 in some instruments, LCM has jazz exams too, but goes up to higher grades.
All of the big 3 exam boards have the same UCAS weighting. IMO no one board is better or harder than another. They just test different things.
TG tends to be more performance based and is much more flexible for the lower grades in being able to chose supporting tests - they offer improvisation for all levels which I think is great for encouraging musiciality. They have a less onerous scale requirement but, for piano at least, also require technical exercises. ABRSM has less flexibility and way, way, way too many scales at higher grades. LCM is less well known but its exams are well structured and in some ways similar to TG.
ABRSM requires grade 5 theory. TG doesn't require any theory exams (but you'd need to know a far bit to pass the aural tests). Not sure about LCM, I believe it needs some theory to go above grade 6.
Could anyone please tell me what different exam boards there are for music and what the difference is between them?
For instance, are some more difficult than others? Are some internationally recognised and others not? Do some do more classical music? Can you swap between them, doing say grade 3 in one and grade 4 in another? Do they all count for points an the UCAS form?
The ones I've heard of are Trinity, ABRSM, and Rockschool.