Can I transfer the SIM card from my broken iPhone 6 to the new one?

Back in May, someone decided to crack my iPhone 6's screen for no apparent reason. The phone has been fine until this morning, my phone fell onto the kitchen floor and the screen completely broke. It's just a black screen so now I obviously have to get a new phone. I was wondering if transferring the SIM card into the new phone will restore all my photos & texts; basically making it the exact same phone. I doubt that everything is on iCloud, I haven't done a back up in a very long time.

First, there are a LOT of places that can replace the screen on an iPhone and it's cheaper and quicker than a new phone. It will take a couple of hours.

Second, if it's the same model phone and the same carrier (ATT, Verizon, Sprint…) you can just swap the SIM.

Yes certainally.

You have to restore the new phone from a backup of the old phone to get everything back. Moving the SIM just moves your telephone number and cell data service.

I don't think so. I think you can still fix your iPhone 6

The SIM card is not a memory card. All of that data would be on the phone. If you haven't done a backup then you would likely lose that data.