What are the advantages of 6 wires LVDT over 5 wires LVDT?
With 5 wire system you save some copper wires on remote sensors.  This matters on aeroplanes more than elsewhere due to weight and not cost.

With 6 wires you can get full accuracy with simpler receiver circuit.  Check the data sheets to see if it matters, a high quality 5 wire system may surpass a cheap 6 wire system.

The current flowing in the common wire should be small but any current flowing in it will result in voltage drops which will shift the differential voltages by a small amount.  You will often find that two wires are joined in your instrument but having the extra conductor to the sensor means the two phase windings do not interfere with each other.

Like 3/4 wire RTD circuits the extra wire can be compensated for with electronics but sometimes the wire is cheaper than the circuitry.  

Also some receivers will only work correctly with one arrangement so you have to match sensor and instrument.