They provide increased control ability around all axis (with software for automatic stabilizing). They are more stable than helicopters due to the appropriate arrangement of the center of gravity and aerodynamical center; with setting up four propellers position (and rotation frequency, if applicable). Therefore, when you install something on the vehicle, change of center of gravity can easily be tolerated by the software.
Most of the helicopters were designed to enjoy manual flying. In fact, if there were a control software, there would still be a possibility to make it unstable with installing something inappropriately. To simplify the concept, I can give the bicycle, and car instances.
The primary reason for the use of quadcopter designs over helicopter for small UAS is mechanical simplicity. Here is an informative article on the subject: http://www.forbes.com/sites/quora/2013/12/23/what-makes-the-quadcopter-design-so-great-for-small-drones/
Abdul, Tevfik and Kevin have covered nearly everything concerned with this question.
Quadcopter or any multirotor configuration are implemented only in small / micro unmanned aircrafts. As already mentioned: in such sizes, masses and volumes a helicopter configuration is too complex and too difficult to implement.
Unlike a helicopter with a single giant rotor that is unsafe to use by a civilian user, a multirotor can be made to be much safer, easier to use and can be set up to accommodate strong / uneven wind and loading conditions as already said by Abdul and Tevfik.
All this is considering a complete electrical flying platform where the only moving parts are the motors. So comparing a purely electro-mechanical system to another with an internal combustion engine is not on the same lines. Also comparing the advantages/disadvantages, savings and efficiency between a quadcopter and a helicopter cannot be done if both are not using electric motors.
However, when it comes to seriously considering aspects like flight endurance, flight range and payload capacities things - like a helicopter that runs on some form of fuel rather than electric motors alone, a quadcopter with geared rotors instead directly fixed on the motors, etc. - will all come into play. Here comparing the 2 will not be possible. Put simply you'll need to address and clearly establish what the application of the aircraft actually is; it will come down to where and how much is the capability of hovering required / utilised.
Besides the already mentioned, I think one of the biggest advantage is the lag of anti-torque requirements (i.e. a tailrotor) which makes the quadrotor much more agile in all directions and allows all available power to be utilized in lifting direction. However, it makes control design also more complex, which in small UAS electric setups is easily achievable via motor/rotor rpm control, but can't be emulated in a similarly direct way in full scale, turbine or recip powered UAS. However, there are comparable helicopter designs with similar trades, usually in a dual rotor (coaxial, tandem, or intermeshing) designs, e.g. the Russian Kamov helicopters with impressible side-way maneuverability and speed...
Do agree with all comments .would like to add regarding solidity which increases on the account of having 4 propellers in quad as compared to helicopter. Solidity effects increasing the swept area as such the efficiency is enhanced.
Simplicity,stability,safety,efficiency and cost are the factors to be considered.
Quadcopters are mechanically simpler and hence, cheaper to make. They are less stable and efficient compared to helicopters. They can only be balanced effectively when all four motors are working satisfactorily.