You can get quite pedantic about listing things I guess - but I had a relaxed relationship with my parents, but in the evenings I'd have to socialise with them, and spend pretty much all my time with them, unless I was going out. I couldn't just invite a bunch of mates round to watch a movie, or have a meal and a few beers - it would feel a bit weird doing that. My parents wouldn't have minded, but you can't really be totally relaxed (things like swearing and so on.. not that I swear much anyway).
I like to be able to decorate and buy the equipment I want, and place them where I want. Heck, even walk to the loo in the buff if I want to. Prepare the foods I want to eat (sure I could do that, but it seems daft/uneconomical to cook 2 different meals). Be able to do what I want on my own terms at any time.
Again, I'm sure depending on your relationships with your parents you can do all of that - but also I can't help but respect my parents "way", as such I constantly felt like a guest at home and not really ever able to relax fully. I also felt embarrassed being at home still in my young 20s - I couldn't imagine being still at home when I was 35!
Ultimately, having your own space and your own rules to do what you want is the ultimate definition of being independent. You're free to do and behave any way you want. Initially you go a little nuts, don't tidy, or buy crap food, or whatever - but after a while you start to get house proud, and get a routine, and learn how the world works better (dealing with councils, home insurance, appliances, trades (gas/elec etc...), and the need for maintenance - all that jazz. That experience is diluted when you live at home, no matter how much responsibility your parents give you.