This paper studies the energy efficiency of the cloud radio access network (C-RAN), specifically focusing on two fundamental and different downlink transmission strategies, namely the data-sharing strategy and the compression strategy. In the data-sharing strategy, the backhaul links connecting the central processor (CP) and the base-stations (BSs) are used to carry user messages -- each user's messages are sent to multiple BSs; the BSs locally form the beamforming vectors then cooperatively transmit the messages to the user. In the compression strategy, the user messages are precoded centrally at the CP, which forwards a compressed version of the analog beamformed signals to the BSs for cooperative transmission. This paper compares the energy efficiencies of the two strategies by formulating an optimization problem of minimizing the total network power consumption subject to user target rate constraints, where the total network power includes the BS transmission power, BS activation power, and load-dependent backhaul power. To tackle the discrete and nonconvex nature of the optimization problems, we utilize the techniques of reweighted $\ell_1$ minimization and successive convex approximation to devise provably convergent algorithms. Our main finding is that both the optimized data-sharing and compression strategies in C-RAN achieve much higher energy efficiency as compared to the non-optimized coordinated multi-point transmission, but their comparative effectiveness in energy saving depends on the user target rate. At low user target rate, data-sharing consumes less total power than compression, however, as the user target rate increases, the backhaul power consumption for data-sharing increases significantly leading to better energy efficiency of compression at the high user rate regime.