So why advocate the bridge circuit over the 'direct' approach I show on the right,
I think it's because the bridge configuration presents a balanced impedance to the inputs of your differential amplifier. Any common mode signal will just be rejected and only the difference (i.e. the change detected by your sensor) will be amplified.
Imagine this, if you have any "external" noise/interference that got picked up, chances are the noise will be picked up by both lines, and therefore should/will be rejected at the inputs of your differential amp. Only the true signal (i.e. the data you want),will be accepted by the diff opamp. So it has better noise immunity.
Addendum:
Here you go, see this... page 2.