On a Mac or Windows desktop/laptop, you have your iTunes program to interface with your mobile device (iPad/iPhone). You also purchase apps and media through the iTunes store when your iOS mobile device is syncing with your desktop/laptop. But once you use your iOS mobile device to make purchases, you have the App Store for purchasing mainly apps and games, and your iTunes store for purchasing music, movies, books, and other media. They made it that way on the mobile device to segregate the apps from the media purchases.
When you ask a veteran Apple customer where to purchase apps and media, he/she'd quickly say, "You can download them on iTunes."
When you ask those who are recent owners of an Apple product, he'd say, "You can download games on the App Store and music on iTunes Store."
To set the record straight:
iTunes - desktop/laptop program for downloading media and mobile apps
iTunes Store - mobile app for downloading media
App Store - mobile app for downloading iOS apps and games
Mac App Store - Mac OS program for downloading apps and games for your Mac OS device(s)
So... is iTunes and App Store the same? From a platform-specific perspective, no. From a functional perspective, yes.