Common mode gain is what you would get if you tied both inputs together and applied a signal to them. The output (ideally) would not be affected.

Differential gain is where you apply a signal so that is appears across the two inputs. This is the normal way, and you expect *huge* variations in the output.

Incidentally, negative feedback can be thought of as providing a small amount of the output to the inverting input so that it is brought to the same voltage as the non-inverting input.

For many practical purposes, it is a good starting point to assume the differential gain is infinite and the common mode gain is zero.

For 99% of simple circuits, that's enough.

You won't generally find the common mode gain specified on datasheets, but you will find the CMRR (or Common Mode Rejection Ratio). This has a mathematical relationship with that gain.

See the

Wikipedia page for the math.

Gain generally becomes important when it is substantially less than infinity.

Of course it's always less than infinity, but if the difference in performance calculated for infinite gain vs that actual gain is insignificantly different, we can ignore it.

It becomes important where the gain required is large, or where the op amp is operating in a region where its gain is reduces (typically at high frequencies). Most often in these cases you're looking at something more complex than a single figure though (you'll find graphs in the datasheets).

Although you might think that finite vs infinite gain is a large issue, often other things are more important. They might include things like slew rate (how fast the output can change), phase shift (affects stability), noise, input offset, and input/output characteristics.