Edit: Oops, should have read the last message. I’ll leave this up anyway, in case anyone had questions about what was going on.
The iPhone can send texts over iMessage (Apple’s service) and SMS (the cell phone service). The iPad (by itself) can only send via iMessages.
SMS works between all cell phones. It is not an internet service, it’s a basic cell service.
iMessage only works between Apple devices.
When you send using an iPhone it is sent via iMessages if the receiving number is an Apple device. If it is not, then it defaults back to SMS. You can tell in the Messages app. When you send/receive from iMessages the text bubble is blue. When you are sending to a non-Apple device the message bubble is green.
The iPad, by itself, can only send to other Apple devices via iMessages, and using an internet connection. However; if youy you have a new enough iPhone and iPad running a recent version of iOS and using the same Apple ID, you can tell the iPhone to let the iPad use it’s SMS service to send messages. To turn this on, go to Settings > Messages > Text Message Forwarding on your iPhone and enable the iPad.