Do I need a common ground wire for differential voltage measurement?
I saw in some cases a common ground wire is needed to prevent common
  mode voltages to saturate the amplifier; but how come a multi-meter
  does not suffer from the same issue?


A multimeter is battery powered and therefore extensively galvanically isolated from true ground.


  Multi-meter can be floating but CM voltages can pass through air via
  capacitive coupling isn't it?


The design of a good multimeter will ensure that capacitive coupling via the hand or bench does not produce an asymmetrical common mode signal to the inputs. That asymmetrical common mode signal would, of course produce a differential error so great care is ensured to make the input circuits as balanced as possible.

It's not that hard to do with a hand held device because a cheap foil shield inside the meter will do the job.


  I'm wondering why in some deferential measurement we need ground but
  in some we don't.


If the measurement device is powered from AC (even via an isolation transformer), there can be significant capacitive coupling back through the transformer and, you might get problematic noise on your measurement equipment ground reference point relative to the device you want to measure. 

Without a ground cross connection, you would have common mode noise on your differential inputs (relative to the signal to be measured) and, inevitably some might be converted to differential noise due to impedance imbalances so, a cross connection between the grounds can prove beneficial but, not if the grounds carry significant earth noise currents due to other equipment present locally in the area.

These earth noise currents can be large (amps) and, due to the cross connection having non-zero impedance (especially at HF), the measurement problem may be made worse. A better solution under these circumstances is to shield both measurement wires and connect that shield only at the measurement earth/ground.

The problems are easy to theorize on but, until the noise is eradicated, the actual nature of the interference can be guesswork.