I have a daughter at Berkhamsted so I know it well. I don't know Haileybury well but my son is at a quite similar boarding school.
I know you already know this but the big difference is that H is a co-ed boarding school and B is a single sex day school (co-ed in sixth form). They do both offer day and boarding places but this is not what they really are, i.e. they are set up differently, with all that means for co-curricular activities, timetabling, bus routes, length of school day, etc. Do not under estimate how much of a difference this makes.
Academically I think B does a good job and co-curricular wise I would say B's strengths are sport and outdoor education. It is the only school I know that has a Deputy Head (Leadership), who is a former Royal Marine Colonel and is just brilliant at his job. The music offer is ok but maybe a bit weak by comparison to other senior schools I know. Family wise (if that matters to you) most B families have both parents working, H families will be a more monied set and on average live further away.
The other thing I will say about B is that it is one of the largest senior schools in the country, this means that both as a parent and child it can feel very "corporate". In one way that is good as they have everything; not only facilities, but a specialist in everything, the best IT systems, the most seamless remote teaching offer (not just Teams lessons), etc. But it also means you can feel like just a number, sometimes the school is criticised for lacking a heart and you'll rarely see the Principal.