A differential amplifier and a differentiator are two completely different circuit blocks.
A differential amplifier has two inputs and one output. It takes the difference between the two inputs, multiplies that by the gain, and makes it the output.
Out = (V1 - V2) ⋅ Gain
In this example, the gain is A/B.
A differentiator takes the derivative of a signal. In other words, its output is proportional to how fast the input is changing.
Note that the gain is not dimensionless, as it is for a normal amplifier. For example, the gain can be the output Volts divided by the input Volts/second, which comes out to units of seconds.
In this example, the gain is proportional to -R1⋅C1.
Why do you think the gain could not expressed as V/V?
I'll assume this is referring to the differentiator, since the gain of the differential amplifier is a voltage divided by a voltage, resulting in a dimensionless value.
For a differentiator, the output is the change in the input. Just dividing the output voltage by the input voltage doesn't yield anything meaningful. For example, you get 0 V out for any steady input voltage. Saying you get 0 V out for 10 V in, but also 0 V out for 3.97 V in (or any other voltage), isn't very useful.
What about Acl=-Aol/(1+Aol*beta) with beta=1/(1+sR1C1)?
Since you didn't define any of your terms, nor the context, it's just meaningless characters.