The single opamp diff amp has terribly different input impedance on the + and - inputs. With R resistors all around, the + in impedance is 2R, the - in is 1R, but the differential input is not the sum, it is also 2R; and the common mode impedance is 1R, which may be considered too low.
The two-opamp diffamp is much more predictable. In that case, the + in Z is 1R, identical to the - in, and the differential input Z is exactly the sum, 2R. Common mode Z is lower though, at R/2. In this particular application, the effect of the diode's dynamic resistance variation will have different action on the + and - side of the diff input. This dynamic balance variation may create problems, in particular regarding the Control Voltage rejection, which is a major problem in a diode VCA.
There's a possibility to tweak the one-opamp diff amp by making the resistors in the - branch (2G+1) greater than the ones in the + branch, but it is not really applicable to the current situation, because the gain reduction causes the loop gain to vary.