Phase arrays have some transmission loss when the beam is steered off directly boresight from its plane, with more losses the more the beam is steered off center. Likewise, its also loses receive gain. When you have a fixed four set, the strength of the radar signal is not even, and there is going to be four weaker zones around the corners of each face. Rotary antenna gives an even signal around a 360 degree field. In fact the beam need not be steered electronically, it can be steered around horizontally with its mechanical rotation.
But that's not the main reason why the rotary is used. The main reason is cost. One face is cheaper than two and two is cheaper than four. Each module in an AESA can cost as much as a small car, and there can be thousands in a single face. If your cost and thermal budget is say, 4000 elements, you have greater range if you put all of them in one single face of 4,000 elements than divide them into four faces of 1000 elements each. Dual face with 2,000 elements each would be between.
The one face or dual face is also lighter, which enables you to put the radar up higher which gives you a longer radar horizon, which makes more advanced warning of low flying sea skimming antiship missiles possible. You can also put a four face up high too, but the total set would need to be light, resulting in smaller faces, so you're back comparing a single face with 4,000 elements vs. dual face with 2,000 elements each, vs. 4 faces with 1,000 elements each.
Of course, the four face has 360 degrees coverage simultaneously, the one face has to turn around for the 360 degrees and the rate of rotation is important. The dual face doubles the rate of rotating updates, sitting between the two.
One disadvantage of the rotary set is mechanical wear and tear, the radar has to be periodically taken down for overhaul. We see this happen on the Type 054A, where the top radar is removed for overhaul.
Another disadvantage is that if the radar has to be put on top of a mast, there is a weight and thermal budget that goes along. The radar cannot be very large, it cannot be very heavy for obvious reasons. The space is limited which limits how much heat it is allowed to generate internally, which curbs its power, and therefore its range.
One advantage which applies to fixed AESAs only is that superstructure and mast mounted fixed AESAs allow backside access for the crew to replace defective modules during the sea. After this, the radar can be recalibrated while on sea --- which can explain the calibration booms you see on the 052C and 052D --- although this system can, should and was replaced with an internal system later. If the AESA is mounted on a rotary, there is no access to it and repairs can only be done on port.
Rotary AESA can deal with supersonic ASM, or at least that is what the advertising for such radars will tell you.
It doesn't need to spin very fast constantly. It can slow down, so the face is allowed to dwell on the target longer. If the radar is turning in one direction, the tracking beam can contra rotate, which also allows a longer dwell time on the target. Remember the array allows an X and Y scan, compared to search radars that only electronically scan vertically, and need to be mechanically turned around for horizontal scan, like the Type 382.