The biggest reason that fly-by-wire hasn't been implemented yet is the complexity of the change required. Larger aircraft already started using hydraulics to move their control surfaces, and a fly-by-wire system adds to this by actuating them with a computer. Smaller aircraft do not have the need for hydraulics, so they must move from direct linkages to computer control. As ratchet freak says, this adds a lot of weight and complexity.
Flaps are much more simple than a whole control system. They don't move very often, and if they fail, the plane is still able to take off and land safely. The pilot directly controls them on a simple scale of up to down. The rest of the flight controls must be able to move throughout the flight, be quick to respond, and implement computer control logic. Failure is much more critical than the flaps as well.
There is also a large inertia of history to overcome. The industry is used to traditional control linkages. Moving away from this will incur a large cost in developing the system, testing it, and getting it approved by regulators. With sales at their current levels, adding this to older designs may not be worth the cost of development, and new designs would have to make the option attractive compared to other options.
A big hurdle to overcome is also safety. While larger aircraft have managed to implement these systems, the challenges are greater with smaller aircraft. Mechanical controls are generally more simple to inspect and repair, and the GA maintenance community would have to learn many new skills to work with the new systems. Redundancy and safe back-up controls would have to be included in a smaller airframe. All of this when regular controls work perfectly fine and are efficient in cost, weight, and complexity.
As reirab commented, stall training is required for a PPL. It makes sense that pilots should still know how to handle a stall, in case the protections fail. But this means the system must have an option to be inhibited. So although the plane should usually prevent stall, pilots will also need to know how to recognize and handle system failures, which even experienced pilots with an ATPL can struggle with.
So what benefits are there? Safety is one area where fly-by-wire can offer great improvements. A fly-by-wire system can automatically prevent the pilot from stalling the plane or otherwise putting the plane in a position where it is out of control. These kinds of incidents are much more common in general aviation than with airliners, so there is a huge potential for reducing the number of accidents. The plane can also trim itself automatically and deal with many of the aerodynamic idiosyncrasies that different small planes have.
There is progress being made here though. Fly-by-wire technology has been used on business jets like the Dassault Falcon 7X. Diamond is working on a fly-by-wire option for the DA-42, as well as the future DA-62. As of 2012 they were hoping to have this available around now, but there are at least manufacturers working on bringing this technology to market.