I've heard from informal publications and from crewmembers that there are two reasons. First, the crew needs everyone's undivided attention during takeoff/landing, should things go suddenly wrong. Second, some have said it causes interference.
Based on the first answer, I've often wondered why it's okay to read a book but it's not okay to read from a tablet or computer. I think airline companies or crews just think it's a slippery slope and would rather ban them entirely than make the rules more complicated.
Many (US) airlines now allow tablets to be used during takeoff/landing provided they are small enough (e.g. less than a certain weight). Check with a specific airline for details.
UPDATE (7 January 2017):
It really depends on the airline and applicable law. The "Mobile phones on aircraft" Wikipedia article has an excellent overview of this. For those of us in the USA, this excerpt from that article may be helpful:
Contrary to popular misconception, the Federal Aviation Administration
(FAA) does not actually prohibit the use of personal electronic
devices (including cell phones) on aircraft. Paragraph (b)(5) of 14
CFR 91.21 leaves it up to the airlines to determine if devices can be
used in flight, allowing use of "Any other portable electronic device
that the operator of the aircraft has determined will not cause
interference with the navigation or communication system of the
aircraft on which it is to be used.
With more people having smartphones, tablets, and the like, and with the increasing popularity of WiFi on airplanes, there appears to be more pressure to allow such devices on aircraft throughout the flight. Still, it appears to be up to the airline to set and enforce their own rules.