Q: Airplay - Different Functionality on Apple TV vs. Airport Express?

I want to be able to wirelessly stream music directly from my new iPad and I also need a new router so the Airport Express is looking to be the device I am after, but I am confused. Both the Apple TV and Airport Express feature "Airplay," but it appears that the functionality of the Airplay feature differs between these two devices.

The description from Apple TV states: "AirPlay lets you wirelessly stream what’s on your iOS device to your HDTV and speakers via Apple TV."

The description from Airport Express states: "AirPlay takes the music from the iTunes library on your computer and sends it wirelessly to any stereo or speakers in your home."

I interpret this as the Apple TV will allow me to stream music directly from my new iPad but the Airport Express does not. The Airport Express will only allow me to stream music directly from the iTunes application installed on my computer and use my iPad as a remote.

Am I interpreting this correctly? With the millions of iOS users out there, this seems so confusing and misleading to call this feature "Airplay" on both devices when the functionality is so different.