Reduced system weight isn't really the biggest advantage.
The flight Control computers recieve many more inputs than just control column/stick position- they recieve flight data such as IAS, ALT, G, AofA etc, and thus tailor the flight control reponse to the conditions, allowing the designer to program the aircrafts response, making the aircraft easier to fly and with more consistent response across the performance envelope.
This allows for things like Envelope Protection, where the control will either not allow, or give vey clear warning, if certain parameters (approach to stall, overspeed, over bank, "G" exceddence) are exceeded.
Another POTENTIAL advantage of FBW is relaxed stability (with "Active" flight controls), the controls providing input to simulate aerodynamic stability. This can lead to reduced drag.