Why aren't cell phones allowed to be used in aircraft?
Why aren't cell phones allowed?/ What problems to cell phones cause?

Most of these cases apparently originate from a series of unexplained events in the 90s which were believe to be caused by electronic gadgets, hence the ban. This was when cell phones appeared, and there was an aggressive ban as a result. NASA has a list which includes some of these and makes for an interesting read. 

Now however an increasing list of airlines is allowing them again, depending on jurisdiction. FAA is working on allowing it from a regulatory standpoint. Bear in mind that phones were probably much more powerful back when these rules were introduced.

Mythbuster (episode 6, 2006) did a test on this by sticking in a powerful transmitter into a business jet. Their website gives a slightly different version: It was the FCC, not the FAA, who led the way for this regulation:


  Explanation: Never mind what the chatterbox in the seat next to you
  says about cell phones messing with plane navigation — those metallic
  birds are built airtight against foreign signals and operate on
  entirely different frequencies than cell phones.
  
  So why all the fuss about phones? When you make a call at 10,000 feet,
  the signal bounces off multiple available cell towers, rather than one
  at a time. That means too many phone-happy jetsetters might clog up
  the networks on the ground, which is why the Federal Communications
  Commission (FCC) — not the Federal Aviation Administration (FAA) — banned
  cell use on planes.


How is using a cell phone in the air different than using it from the ground?

Is it sensible to allow them on aircraft? To quote one of the above sources:


  although the FAA recommends that you still switch to airplane mode
  because you’re not going to get a signal 30,000 feet in the air — the
  only hit you’ll take is a dead battery when you land.


So it's probably not a huge loss if you're flying commercial aircraft at 10,000 meters. In any case, you can't expect a service like you do on the ground, unless like Lnafziger says that the aircraft is equipped with a cellular basestation called a Picocell, illustrated below:


(source: wordpress.com)