In addition to what tdracer and gums have put forth above as reasons for FBW on commercial airplanes are the performance benefits that FBW supports. It is certainly the case that FBW heavy transports must be designed to support continued safe flight and landing if system failures lead to use of simplified, backup control laws. The handling qualities that are achieved with those backup laws would, however, not be certifiable for every day usage. The open loop aero characteristics plus the limited amount of augmentation provided in backup modes do not have to be good enough for regular usage. They are allowed to be Level 2 for those familiar with handling qualities rating scales. This is permitted because of the high reliability and availability of the full-up up normal systems that produce desired response characteristics.
Relaxing the requirements on the inherent aerodynamic handling qualities allows for designs that employ relaxed static stability and high performance wings that do not present desired stall characteristics on their own. Relaxed static stability permits loading further aft such that trim drag is reduced. This also allows reduction in the horizontal tail area as it does not need to generate as much down force to balance wing lift and CG location. Look at the ratio of horizontal tail area to wing area across a number of models: 737-200 (28.4%), 737-800 (26.3%), 747-200 (26.7%), 787 (23.8%). A smaller horizontal tail is one measure of static stability. The 787 has a comparatively smaller tail and along with that lower trim drag thus better fuel economy. Being able to rely on augmented FBW to provide desired stall characteristics allows use of thinner, higher aspect ratio wings that again have improved trim drag performance.
Another added benefit of augmented FBW is envelope protection and load alleviation functions that further increase performance by enabling lighter weight designs. Those are two large stories on their own that I will not dive into with this posting.