My iPhone does not do this, and yours should not. Unless you shut all updating and backups down, LTE would get hammered if the iPhone did not continue to use wi-fi, even when the screen is off.
Or it will simply fail to do any iCloud backups or app updates in the background.
If you are certain your iPhone is using LTE data whenever the screen is off, I recommend contacting Apple and asking what's up. It should not do this. If you have Applecare, you should have free tech support. Or you can wait until you have the iPad, and if it does the same thing you've got 90 days of free tech support for it.
Or it might be a problem with your router killing the connection. Restart it, Reset All Network Settings on the iPhone, connect back to the router and see if the problem goes away. Also, check with your manufacturer's support page to see if there are any firmware updates for the router.
Basically, LTE opens up all kinds of searching, browsing, and location tracking while you are on the go. You can also take advantage of the various streaming services (while keeping a close eye on your data limit).
Whatever you could do with your iPad at home, you'll now be able to do anywhere. Well, anywhere with a decent cell signal.