LM324 Difference / Instrmentation Amp 1mV Signals
This is a discrete implementation of a classic 3 amplifier Instrumentation amplifier, where the key specifications are Common Mode Rejection Ratio, Input Offset Voltage, input offset current and effective resistor match. Other specifications may be important in a particular application, but these are the dominant issues for standard instrumentation amplifier applications.

The common mode rejection ratio is required to be high particularly at the DC offset from the bridge (usually Vcc/2, and that is one reason that many amplifier parameters are primarily measured at this input voltage) so that small deviations between the two inputs are properly amplified. The input offset current translates into an effective input offset voltage due to source impedance imbalance.

The input offset voltage must be very low, as this is amplified by the gain of the amplifier into an output offset voltage. At a gain of 470, the LM324 typical Vos will yield an output offset of almost 1V (and at worst case over 4V).

The common mode rejection ratio of the amplifier is dominated by the resistor match and therefore needs to be closely controlled.

In this case, the amplifier is simply not suitable for this application. You should use a proper instrumentation amplifier (or a 3 amplifier configuration with suitable specifications).

The INA series from TI (originally Burr-Brown) are excellent choices for this type of application, as are offerings from Analog Devices, Linear Technology, Maxim and other TI parts (most from the National Semiconductor range they acquired).

I have not read the specifications for the other amplifier you specified (OP177), but hopefully this information will help you to evaluate the suitability of the device in this application.

HTH