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The 3 op-amp design has three main advantages over a single op-amp differential amplifier.

1. The input impedance is much higher, since the inputs drive directly into an op-amp input rather than into a resistive divider.
2. The gain can be set by changing a single resistor, so the critical parts can be easily integrated on to one chip (maximizing symmetry) with a single external resistor for setting the gain.
3. In high gain configurations the common mode rejection is much better because the gain of the first stage effectively multiplies the common mode rejection of the second stage.

Note that in general it is better to use a specific instrumentation amplifier chip than to try to build it yourself out of separate parts. Having everything on one chip improves symmetry and hence common mode rejection.