Difference between two integrator circuits
First, note that the feedback and input pins are equivalent, they both have a resistive input into the summing node of the opamp. So whatever the opamp does, it does to the scaled sum of the input and the feedback signals, the scaling being set by the respective resistors.

Neither of these circuits is an integrator. An integrator would have a simple capacitor feedback without R2 or R6. However, for much of the frequency range, their response is mostly integrator-like, falling by 20dB per octave, with a 90 degree phase shift, when the capacitor is dominating the response.

So when is the capacitor not dominating the response? When the impedance is dominated by the resistor. 

For the lefthand circuit, that's at high frequency. The gain will stop falling, and become flat above frequencies where C1 impedance falls below R2. As C1 cannot conduct DC, there needs to be some other path for DC feedback from output to input so that the output doesn't ramp off to a rail.

For the righthand circuit, it's at low frequency. The gain will stop rising and become flat below frequencies where C2 impedance rises above R6. Note this includes DC. This configuration is often used down to a very low frequency, with DC stability is ensured by R6.