Why are two stages used for an instrumentation amplifier?

One of the biggest benefits of the 3 op amp INA is the equal and high input impedance. The op amp's non-inverting pins' input impedance can be up in the \$T\Omega\$ range. I'll leave it as an exercise for you, but if you look at the difference amplifier circuit, the input impedance of the negative input varies with the positive input.

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 (highly symmetric) first stage effectively multiplies the common mode rejection of the (less symmetric) 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.

In addition to input impedance concerns, gain in two stages offers better frequency response.