Wow – I didn't even know gatorade had a protein recovery drink… shows how little I pay attention to commercial drinks. Can you get that in any convenient store? Might be handy in an emergency…
Anyway, milk contains both whey and casein proteins, which is one reason it potentially is a great recovery drink (whey acts fast, casein acts over a longer period, there are natural carbs in milk (lactose), and some people strongly swear by chocolate milk (sucrose) for a better carb blend for recovery).
I think milk is like 1/3 whey and 2/3 casein but I might have that reversed.
Keep in mind, other people suggest that skim milk might have some other issues that detract from it as a recovery drink (I'm not sure I really understand their arguments, but it comes from people I respect so I'm not dismissing it). Hence, commercial recovery drinks – some of them at least – really are better for recovery than straight milk.