We normally do self-catered villa in Europe in the summer (actually we normally do an enormous group holiday for one week, and one week just the 4 of us, which will sometimes be hotels as we may be travelling around).
The world is your oyster! As a starting point, what are your kids studying? Languages, History, Classics? Are they sporty, need lots of activity, or happy to swim in the pool and mooch around towns/sights? We try to balance out a week with everyone's input, so everyone gets to do something they want (eg DH - cycling or a sporting event of some kind, me - history/culture, DS - a water park, DD - late mornings, maybe a bit of culture).
As a PP mentioned, temps can vary across Europe, northern France will be similar to UK, but Southern Europe will be baking hot, which can impact on the activities you want to do (Pompeii in August was very, very warm!).
A hotel can give you the sense of a proper break, someone else cooking and cleaning for you, but I find eating out for every single meal gets a bit tiresome, and I yearn for some plain home-cooked food and a simple lunch. I also don't love sharing my space with others. I like a villla with my own pool that I don't have to share. We still tend to eat out at least once a day, but it makes for simpler meals back at the villa (and I love a mooch around a foreign supermarket!), and also means I don't come home feeling bloated and unhealthy!!
For villa bookings, I tend to use VRBO as my first port of call, but AirBNB is also useful.