It’s been a while since I’ve flown anywhere, but it used to be that most airlines would prohibit phone use during takeoff and landing, but would allow phones to be used in flight mode while the plane was airborne.
Also, the airlines didn’t make a distinction between phones and tablets, and just said to turn off all electronic devices during takeoff and landing. Once we were airborne, they would announce that electronic devices could be used, and if it was something like a cell phone or a tablet that it could only be used in airplane mode.
There is a bit of a safety issue, but it’s complicated. First of all, cell phone users are generally stubbornly stupid. It often seems like the best way to get someone to use a cell phone in a particular area is to stick a sign there saying that their use in that area was prohibited. When the public has this vague idea that cell phones can crash the plane, they go ahead and use their cell phones on the plane anyway. So everyone has just had to cope.
Cell phones can interfere with plane communications (i.e. the pilots talking to the tower) and can interfere with the plane’s instruments. I know of one case (prior to 9/11) where a cell phone user completely prevented the pilots from talking to the control tower, but this was over in Europe somewhere. The cell phone user refused to hang up and was arrested upon landing. It’s much more common that the cell phone just causes annoying noise in the pilot’s headsets. Cell phones can also make the compass go wonky and things like that. I’ve talked to a few pilots though, and they mostly say that pilots aren’t idiots. If the compass suddenly turns 90 degrees and the plane hasn’t changed direction, the pilot isn’t going to stupidly follow the compass and fly the plane into a mountain side. Pilots have multiple instruments, and can handle it if one of them starts misbehaving. Similarly, if they do have trouble communicating with the tower, they just ask the tower to repeat what they said. Still, it’s a bit of a safety issue as it’s kinda similar to driving a car while your kids are arguing in the back seat. Yeah, you’re probably not going to crash the car over it, but still you’d like to avoid it if possible.
Plane manufacturers have also gotten good at protecting the plane’s systems from cell phone interference. The electronic shielding on modern airliners is pretty darn good. The wonky instruments is mostly something you see on older and smaller planes, like your typical single-engine Cessna.
The controls for many airliners are also hydraulic. Those systems aren’t affected by cell phone interference. What concerns me is that a lot of the newer planes are fly-by-wire, and while those are intentionally hardened against cell phone interference, you can’t possibly test every single cell phone type against every possible location and use throughout the aircraft. So for engineers like me, the most we can say is that a cell phone possibly interfering with the flight controls may not be likely, but it isn’t completely impossible. There’s no practical way to test to prove that it cannot happen. This makes some people (like me) a bit nervous about fly-by-wire planes.
Some of the cell phone regulations come from the FCC as well. Cell phones in the early days used to wreak havoc on cell phone systems when used in a plane, because not only could the phone reach multiple towers, but the phone could also reach multiple systems simultaneously. This was something that the systems hadn’t been designed to handle, and it resulted in all kinds of problems, including occasionally crashing the systems. Again, because cell phone users are stubbornly stupid, the cell phone companies just had to cope with the problem. They made the systems much more robust, and this sort of thing hasn’t been a problem now for a very long time. The FCC still doesn’t like airborne cell phones, simply because they tie up communication channels in multiple towers simultaneously, which wastes bandwidth. Basically, if your airborne phone is tying up resources on 5 different towers, that’s 5 ground-based users who potentially can’t get a channel to talk on. Most towers aren’t used to capacity, so it’s generally not a major problem.