We both home educate and send the kids to school, they are normally at home for about a third to half the year, and the rest of the time they are at school (this is to do with our professional obligations, not our educational choices), I feel both have their advantages, but that is because my kids fit in school very well, are happy there and have lots of friends, this is not true for all children.
Academically, I find they do much better at home with us, particularly my eldest. In school he fools around quite a lot and is much more interested in his friends than in his lessons. At home he concentrates better and has more choice about what to do and therefore progresses much faster. the HE periods also allow much more scope for extracurricular activities because we are not constrained by the school routine (also, we travel a lot more in those periods), so in general, those period are culturally richer.
The socialisation process is also very different, school has the advantage of providing time with a group of friends on a regular, daily basis, which is nice IF the child has a good circle of friends (it is dreadful if they are bullied, no escape!!), the socialisation during HE is much more varied, freer but less constant and intense, both can be good.
I also like the fact that my kids experience what it means to have to follow rules and fit in, but also get a period when they don't have to do that (well, apart form home rules, of course!), I hope this will give them a good critical sense and distance but also the ability to fit into "the system" when they have to, and like or not, they will have to at some point! but hopefully the system won't crash them if they can reflect about it critically and realise that not fitting in doesn't necessarily mean that there is something wrong with them, if you see what I mean. I feel like we get the best of both worlds and I am very happy with this set up (and realise we are very privileged!).