They do indeed compete but doesn't Apple have a monopoly on IOS with it's app store and Google on Android with PlayStore?

At least on Android you can install you own market if you're so inclined. I have yet to find an alternative app store for ios though(might be just me not knowing where to look though).

Edit: Forgot to add that when you're talking about Apple vs G I don't see it as 1 big "smartphone" market. It's 2 different ones. Apple hardware running iOS vs generic hardware running Android.