Intrinsic gain vs. voltage gain
As I understand it, in e.g. a transistor amplifier, the intrinsic gain will refer to the transistor beta, and the voltage gain to the actual Vout/Vin.

Usually for e.g. a common emitter setup, you would not rely on the unpredictable current gain of the transistor, and add some feedback in the form of an emitter resistor. Then the voltage gain becomes roughly the ratio of the emitter and collector resistors (unless you bypass the emitter resistor for increased small signal gain)
So the intrinsic gain of the transistor might be 200, but (with an e.g. 10k/1k ratio) the actual voltage gain of the circuit ~10.

Similarly with an opamp open loop gain, you could say this is the intrinsic gain of the opamp.

Here are a couple of related links:
MOSFET intrinsic gain
Opamp discussion (see page 3 second paragraph from bottom)