With iOS 6, the iPad can receive iMessages (not SMS) that are sent to an iPhone's number, if they both use the same iCloud account. You can change what email addresses and whether the iPad will receive the iPhone's iMessages under Settings > Messages > Send & Receive. Just check/uncheck what you want, or you can add another email address.
If you no longer use the same iCloud account on both devices, tap the Apple ID at the top and sign out, then sign back in with the new iCloud Apple ID. Do this on both devices so that the iMessages server updates all the phone number and address information.
The difference between Messages, iMessages, and SMS:
Messages is the app.
iMessages's is Apple's text service. I can only be used when you have an internet connection and only works between Apple devices.
SMS is the telephone carrier's text service. It can be used whenever you have a phone signal and works with pretty much any phone (provided it has SMS service).
The iPhone can use either, and will if you have it set up to do so. By preference it uses iMessages when it can (Apple device and internet available), but will switch to SMS automatically; again, when set up that way.
The iPad can only send and receive iMessages; which will explain why some things you get in Messages on the iPhone don't show up on the iPad. They came in by SMS.