These parts are designed for situations where you have a high common mode voltage, such as the example 200VDC.
The AD629 is a a differential amplifier (not an instrumentation amplifier) so the input impedance is relatively low. In the case of a 2K shunt, the common mode rejection is compromised by an imbalance of 2K/380K or about 0.5% so a 200V common mode voltage leads to a 1V error in the output.
There is a current of about 200V/380K flowing from both sides of the shunt resistor (about 520uA) into the amplifier inputs, so your supply is providing an extra mA or so just to feed the amplifier inputs.
An instrumentation amplifier may be better, however you have to keep the inputs within the common mode voltage range of the instrumentation amplifier, which is typically less than (sometimes much less than) the supply rails.
If you are measuring a small current with a small common mode voltage then the AD629 is probably inappropriate. It would be better, if possible, to measure the current on the low side, which requires only an op-amp and does not involve trying to reject a high common mode voltage, with the resulting extreme sensitivity to resistor ratios, which leads to drift and error.
If you provide your own resistor dividers on the inputs to an instrumentation amplifier then you have to consider the stability of the ratios and the increased noise of the (already usually noisy) instrumentation amplifier. And you still have the effect of the shunt resistor imbalancing the ratios.