iTunes is an application. The dock icon looks like this:
This has the iOS app store, iTunes U, and many other Music related things. The iOS app store is for purchasing apps solely for iOS devices like iPad or iPhone.
App Store, or Mac App Store, which is available in 10.6.6 (Snow Leopard) and and later release.
This is for buying apps for your computer.
Any purchases made in the (Mac) App Store are also available on any other computer that you use the same iTunes account with for free.
Any purchases made in the iOS App Store, or the iTunes store are also available on any other computer or iOS device that you use the same iTunes account with for free.
a bit clear now.
So the App Store icon on my ipad it´s the same store inside Itunes (called Itunes U). But it´s not the same App Store that i have on my macbook.
It would be better if they have just one site, same name, with the content related to the device you are entering from...
Not to say that sync apps with Itunes is a pain. The less intuitive thing i found in computers for years.