Why can't planes use only flaps in the tail section instead of stabilizers?
Your question is partly covered here regarding why the vertical stabilizer is important. The empennage (tail surfaces) of conventional planes provides both passive and active stability. Like the feathers on an arrow (which is where the word empennage comes from), they help the airplane to remain pointed in the direction of flight as well as providing active control surfaces. The cost of providing stability is added weight and drag, but if the surfaces are removed, the stability aspect will need to be provided manually, or alternately. This is how unconventional aircraft like the B-2 flying wing, or even modern fighter aircraft, are able to fly.

So, yes, we can fly airplanes without conventional tail surfaces. Some designers like Burt Rutan have created unconventional designs for general aviation. However, design changes will also have drawbacks. At the very least these designs behave differently than conventional planes, which requires special attention. The strange appearance is also reason for some pilots to distrust them.

If the plane is going to be unstable without computer control, that introduces safety problems. In military aircraft the pilots can eject when issues arise. But unlike modern commercial planes, the pilots cannot simply take over control if the computers fail.

There is also an aerodynamic issue with what you propose. Conventional control surfaces deflect the entire surface, so air continues to flow past both sides. However, if you only deflect one side of the surface, air is only flowing past the outer face. This acts like a spoiler and creates extra drag. Though this drag will be the force controlling the plane, it is not as efficient as moving a control surface. As Peter Kämpf pointed out, it is much more feasible to increase the effectiveness of the existing surfaces than to eliminate them.

Modern airliners represent a compromise between efficiency and safety based on currently available technology. Although alternate designs have been proposed, these configurations create their own unique problems.