Am I allowed to use my smartphone or laptop in the U.S Military?
Yes. It's not prison (although sometimes it feels like it). You're a normal person that goes to "work" at a military base. People in the military are actually probably on their smartphones more than the average civilian at work. When you're a lower enlisted Soldier in a combat job (but not deployed), you spend a lot of time sitting around. Honestly, being in a Cav Scout/Infantry unit in garrison (not deployed) is incredibly boring. You're either training in the field, cleaning, or sitting around waiting for something to do.
The caveat to that question is if you're in basic training. In basic training you will not be allowed to have either but you will get your phone back occasionally to make calls to family.