BJT differential amplifier
The single ended circuit did not have the AC voltage source on the
  left hand side


When you apply no AC signal to Q1's base, the shared emitter connection is still largely held at a small signal value and the gain of the right-side input to the left-side output is the same formula as the full differential mode amplifier gain: -

$$\dfrac{R_C}{2\cdot r_E}$$

The "2" factor in the above equation is because we are applying a differential signal. This means that the voltage gain is halved because each base only receives 50% of the full differential signal.

When applying a single ended signal to a differential amplifier the gain is twice as high but given we might only be applying half the full differential signal, it becomes the same formula.