In a real-world op-amp, how does it make sure Vn and Vp differs and don't equal each other?
A very simple model for an opamp is used in the example below. A voltage-controlled voltage source with a gain of 1000 is used to build a non-inverting amplifier.



As you can see a difference of about 2mV is needed to get a voltage of about 2V at the output (since the gain is 1000). For a smaller difference at the input the output voltage is reduced and the voltage at the inverting input gets smaller. So it's actually the external negative feedback that takes care of a non-zero differential voltage and not the opamp itself. It just provides high gain.