there is a lot going on here. Short answer is that a ducted fan (what you have pictured) can produce a lot more thrust (experiments from one paper say twice as much) than an open rotor of the same size. The big thing that these ducted fans have going for them is the fact that tip losses become negligible because there simply isn't room for the tip vortex to develop because there is a duct in the way! Simple, easy solution that works well.
However, nothing comes for free. One, you have to have the duct there, which is weight & money (you have to build the duct and the support structure, which adds some drag too). Think of scale too--as propellers get larger, the duct has to grow with them. Imagine the difficulties in putting a 20' duct on a B-36 propeller!
Second, the manufacturing tolerances on the duct are critical--the tighter it is, the more efficient you are (to a limit, naturally)...but the tighter it is, the more susceptible you are to having your blades strike the duct due to flexing of the duct or the blades. And, speaking of flexing blades and the necessary support structure you'd need, this could make pitch control (on a propeller, variable pitch, and on a rotor, cyclic and collective) difficult to implement, depending on which of those schemes you're looking to do. Then again, for a small implementation using RPM control...maybe you don't need those anyhow.
Similarly, while a crosswind on an open rotor leads to a sudden decrease in power, a crosswind on a ducted fan leads to a sudden pitching moment due to the lip of the duct and the asymmetric inflow to the fan (which, since you're now producing more thrust to begin with, asymmetric inflow effects become even more pronounced). Now, if you stick this on the front of a typical Cessna 172, say, and throw a crosswind at it (which, if you think of takeoffs and landings, which since they must be in a certain direction, having a crosswind is actually quite likely) imagine the yawing moments that would be induced by the propeller (proportional to the static thrust) -- even before the aircraft got off the ground and before you have a lot of control authority to counteract it! A similar scenario in the air could be as dramatic. And, yeah, you could throw some guide vanes at it to help, but nothing comes for free. Guide vanes, which is a workable solution, now mean that you need to have actuator(s) for those vanes, power & maybe hydraulics for the actuator(s), mounts for the actuators, hinges for the vanes, linkages from the actuator(s) to the vanes, redundancy for the actuation or some means of facilitating control in an emergency...it's a lot of work. Or, by contrast, you could just bolt a fixed pitch prop on a shaft at the front of said Cessna 172 and call it a day.
In the long run, and this is some speculation, I think it just leads to a more complex system, particularly in full-size implementations, which is a maintenance, cost, and manufacturing headache that isn't worth having if you can get plenty of performance from a regular open propeller. If you venture into the world of diffuser augmented wind turbines, there have been similar debates going on for decades...and a ducted wind turbine has yet to be successful on the large scale. Similarly, while ducted fan aircraft have flown...
(Ryan XV-5A Vertifan, circa 1950. The ducted fans are underneath the circular domes on the wings and another under the shuttered area on the nose for pitch control. Source: http://www.aero-web.org/specs/ryan/xv-5a.htm)
...there aren't any non-MAV examples around today that I know of.
Source:
https://theses.lib.vt.edu/theses/available/etd-05262005-170916/unrestricted/WGraf_Thesis_2005.pdf
