What are the pros and cons of a pulsejet?
The pulse jet engine works because it uses the dynamic pressure of air for compression. Consequently, it needs speed to work well. If the pulse jet is on the ground, it is not easy to start - it needs to be fed with compressed air, and once it runs (it can run even when not moving because the oscillating pressure wave inside the tube will compress the inflowing air), it develops litte thrust.

Thrust will go up with airspeed, so you need additional thrust sources for initial acceleration and take-off. In case of the V-1, this was done by a steam catapult or by air-launching the device. Note that the V-1 was accelerated to 370 km/h on a 45 m long catapult which means more than 100 m/s² (10g) average acceleration. Most of the development work went into making the airframe withstand this acceleration. A modern design would need additional engines as well, and once those are included, the rationale for pulse jets becomes very weak. They make sense if you have an air-launched, disposable application, such as long-range air-to-air missiles. This is a niche in which their higher-speed cousins, the ramjets, will still be used.

Pros:


Very easy to build, very light.
Runs and produces some thrust when at rest (as opposed to ramjets). 


Cons:


Thrust grows with flight speed. This can be seen as a pro, but normally requires additional means of acceleration. Therefore listed as contra here.
Low fuel efficiency. The fuel energy is converted to other forms of energy: Just listen to the videos of Colin Furze to get an idea.
High levels of vibration due to the intermittent operation. When the Heinkel 280 was tried out with Argus pulsejets, the aircraft experienced unacceptable vibration.
Need heat-resistant materials. The central part will get very hot in operation, so at least nickel-chrome stainless steel is needed if air cooling is insufficient.
Cannot go supersonic. For this, there are ramjets.




EDIT:

NACA tested the As 014 pulse jet and wrote a report, from which the performance data depicted below is taken:



It seems that Bruce Simpson over in New Zealand has improved the pulse jet design a lot: He needs no compressed air for starting and makes clever use of surrounding cold air to increase mass flow and keep material temperatures down.