Hello there,
I think the simplest way to distinguish these differences is as follows...
The transfer function for a perfect integrator is A/s, while for a first order low pass filter it is A/(s+B). Clearly, the difference is in the denominator with the constant 'B'. Now if B is small it will approximate A/s so we can say although it is not a perfect integrator it resembles one. In other words, when B is small A/(s+B) approximates A/s.
In the practical case with a given value for the feedback resistor, you can calculate B and see the difference in B when using a high value resistor and when using a low value resistor.