Existing works concluded that coherent transmission outperforms non-coherent transmission in the downlink of cell-free systems when the fronthaul links have unlimited capacity. Since the capacity of the fronthaul links of cell-free networks is typically limited, in this paper we ask the question whether this conclusion holds under more realistic assumptions on the fronthaul capacity. To answer this question, we study and compare the performance of these transmission strategies by formulating novel energy efficiency (EE) maximization problems for both strategies, where we explicitly consider realistic fronthaul capacity and power consumption constraints. Despite the non-convexity of these problems, we derive closed-form equations to find suboptimal solutions of both problems using a unified framework that combines successive convex approximation and the Dinkelbach algorithm. Numerical results show that the performance of coherent transmission is severely impacted by limited fronthaul capacities, power consumption on the fronthaul links, user-centric cluster size and the number of antennas at the access points, such that in many cases non-coherent transmission achieves higher EE than coherent transmission. Based on these results, we provide deployment guidelines on when to use coherent or non-coherent transmission to maximize the EE of cell-free systems with limited fronthauls.