What are the advantages and disadvantages of autogyro aircraft?
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From the same website:


  Zooming through the sky in something that looks like a flying chair, isn't for the faint-hearted.
  
  These aircraft look like a cross between a small helicopter and an upside-down lawnmower.


Autogiro's are used by bush pilots for mustering in Australia, due to their versatility and relatively low cost. And are also used for police patrolling in Tomball Tx.

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Advantages of the autogiro:


Versatile and small.
It can fly very slowly, safely. From this site:


  Gyros cannot stall because the rotational speed of their rotors, and hence their lift, is not dependent on forward speed. At zero forward airspeed a gyro will slowly descend, to a survivable landing. 

It can land anywhere.
It can takeoff from almost anywhere if the jump start is used: the engine spools up the horizontal rotor, then after take-off drives the propeller only.
Less complicated than helicopters, resulting in lower acquisition cost - much less complicated rotor hub.
Lower running costs than helicopters.


The disadvantages:


They are classified as recreational aircraft, with implications on licensing and oversight.
Some statistics show them to be more dangerous than most recreational aircraft. From this website:


  Aviation statistics show gyrocopters to be more dangerous than most recreational aircraft. An Australian Transport Safety Bureau report on accidents between 2004 and 2013 found half of all gyrocopter accidents to be fatal, with the aircraft responsible for the worst fatality rate per hours flown.


This can also be because non-fatal accidents with autogiros are under-reported.
Licensing (in Australia) is done through the sports association, which implies no access to commercial use of gyrocopters.


The simple autogyros shown above are not built for speed with their un-aerodynamic shape. Power to overcome parasitic drag scales with (engine power)$^3$, and aircraft designed for high speed have a much more streamlined shape than the gyrocopter has. From this site:


  Gyroplanes have somewhat different flying and performance characteristics to fixed wing aircraft. They have high drag and are therefore relatively slow, with typical cruise speeds of between 50 and 60 kt. The gyro’s advantage is not top speed but its ability to fly slowly, which is why under the name autogyro they were developed in the 1930s as a precursor to the helicopter.


But the flying armchairs for sheep mustering are not the only types of autogiro's. The fastest helicopters have a vertical thrust propeller as well:

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Technically not an autogiro because the main rotors are driven, but the thrust propeller helps relieving the main rotor forward thrust. The Fairey Rotodyne was a pure autogiro, aerodynamically shaped, and had a maximum speed of 166 kts. Further speed increase is only possible when applying coaxial rotors like the X-2, to remedy instability due to retreating blade stall.

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