Why Op amp amplifier's voltage difference zero?
The main thing that is almost always mis-stated or mis-understood in all these examples is the fact that the inputs being close or even identical to each other is NOT a effect of the amplifier but is an effect of negative feedback.  The amplifier is assumed to be ideal to simplify the discussion.

An ideal op-amp without negative feedback will NOT have the inputs being identical.

What your professor means with the input impedance statement is that if there is a measurable input impedance thane there will be a potential difference between the pins and the actual differential element inside the chip. this manifests itself as an offset voltage of one input vs. the other, which gets amplified and manifests itself as an offset on the output.