Any app that exists will download into iTunes on the computer and be shown in your app library, because it has no idea what device you're going to install the app on. Sure, it knows you have an iPad, because you've synced it before, but it's not going to assume you don't have (or plan on having) a new iPhone to use the app on.
By default the App Store on the iPad will only show you apps made for the iPad. You can see apps for the iPhone that will also work on the iPad, if you change the search criteria to iPhone Only, but even then it won't show you iPhone apps that the developer did not explicitly say would work on the iPad.
On the iPhone, you can not see iPad apps at all, because they won't work on the iPhone, at all.
The four types of apps:
1. iPad only apps. They work nowhere else.
2. iPhone apps that will work on the iPad, but show an iPhone interface. If you read the Information section in the App Store it will say it will work with the iPad, but there are no iPad screens shots; and it won't show up in the iPad's App Store unless you search for iPhone Only.
3. iPhone apps that for one reason or another won't work on the iPad. Usually they need a phone number or other hardware specific feature. These don't work on the iPod Touch either, for pretty much the same reasons.
4. Universal apps. Denoted by a + sign in the purchase button. These work on both devices, switching the the best interface for the device you're using. They show up on both devices, and in both the iPhone Only and iPad Only search modes.