Internet won't work but still receive iMessage.
There are two things that you need to pay attention:  

First: Having Cellular Data ON allows a user to select apps that can access the internet via a cellular connection.
So first check whether the app with which you are trying to access the internet is allowed to use Cellular Data or not. The list of apps can be found here:
Settings > Mobile Data > Use Mobile Data For:

Second: iMessage sends messages via Cellular Network when Cellular Data or Wifi is not available. So maybe you are getting your messages via cellular network and you are not connected to the internet even though your Cellular Data is ON.