The Messages app on the iPhone is different than the Messages app on the iPad.
The iPad app can only use Apple's iMessage service; which is only good for sending messages to other Apple devices/accounts., and needs an internet connection.
The iPhone app can use both the iMessage service and SMS. It will use iMessage by default, but if the phone number you are sending to is not another iPhone it will use SMS; which is available on the majority of cellphones.
SMS is part of the older cellphone technology and only works if you have an actual dial-able phone number. Since the iPad can not make phone calls, it also can not send SMS. Your iPhone does not need internet to send SMS, only a cellphone signal.
There are workarounds. Most of them involve an app and service combination that supplies you a phone number on the iPad to both send and receive SMS. The app store is full of them. Hold in mind most cost money, or have limited service until you do pay.