Hi,
Maybe he is having trouble understanding what a differential input is?
A differential input is a set of inputs that detect the difference between the two inputs and generate a signal based on that difference rather than on the absolute level of some single signal. So it gets two signals then subtracts them, and uses the result of that subtraction to generate an output.
The types of inputs can be either voltage or current, and the type of output can also be either voltage or current. It depends on the amplifier design.
With an OTA however, the inputs are voltages and the output is current. That and the differential input means that it will measure two voltages, subtract them, then amplify that difference and output it as a current level. The internal gain of these amplifiers can be high but we can also adjust it with the additional terminal.
For example, say we have an OTA with internal gain of 10, and on the input inverting terminal Vn we have a DC voltage level (measured from ground) of 2.000 volts, and on the non inverting terminal Vp we have a voltage level of 2.001 volts. Internally the OTA will subtract the two Vp-Vn and so it will do 2.001-2.000 and end up with 0.001 volts, then amplitify that by the internal gain of 10: 0.001*10 to get 0.010 volts but then will output that as a current 0.010 amps. Or we can look at it as 2.001-2.000=0.001 and then change that into a current, 0.001 amps, then amplify that to get 0.010 amps at the output.
That's the basic behavioral operation, but we also see an additional terminal on the real life devices that allows us to change the gain. This leads to some novel applications. In the above example this means we could vary the gain from say 10 to 5, so instead of getting 10ma output we would have gotten 5ma output.
The term "transconductance" itself simply means taking a voltage as input and turning it into a current, usually with some gain. The gain is usually indicated as "gm". So in the above example we had a "gm" of 10.