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General cause of noise/interference
The interference can be distant (such as lightning) or much closer (such as cables connecting "other" equipment) but, whatever the source, interference can be regarded as "dosing" the same energy onto both sensor wires (whether screened or un-screened).
Non-differential input
How does the non-differential amplifier cope with with interference: -
With a single-ended receiving amplifier, one input is firmly tied to local ground and will be largely unaffected by the interference on the cable. Sure, there'll be an injection of energy but, that energy will be "soaked-up" by the local ground producing very little surge artefacts.
However, the other input has an impedance to local ground governed by the input resistor, $R$ and, it will develop a surge voltage that will be amplified by the single-ended amplifier.
Differential input
This problem can be cancelled by using a differential amplifier: -
Because both inputs into the differential receiving amplifier have the same impedance to local ground, the interference waveform is seen equally at both input nodes and, the effect of the differential amplifier is to cancel those signals out.
The other two resistors need to have the same values but they do not need to be the same values as the input resistors. This keeps the amplifier mathematically "differential".
Battery-powered receiving amplifier
You might consider that a battery powered receiver might not need a differential amplifier because the interference would affect both inputs of a single-ended amplifier similarly like this: -
The above would be OK if the "floating 0 volts" node had the same capacitance to real ground as the node connected to resistor R. This will hardly ever be the case and inevitably, you will still get a differential noise voltage like this: -
Using a differential input amplifier is effective because it "presents" a balanced impedance to ground for both wires in the cable.
Source/sensor grounded
The case for a differential amplifier has hopefully been established and, this is also true when the sensor is grounded but now, the output impedance (X) of the sensor has to be factored in: -
Because the sensor has a non-zero output impedance (X), cable interference will not be "handled" equally on both wires despite the receiver being differential and balanced. To counter this, you make both wires connected to the sensor terminate with the same impedance (X): -
In the above, both wires will present the same impedance to ground and this is called: -
$$\boxed{\text{A balanced impedance transmission system}}$$
And note that a balanced transmission does not require a differential-output sensor; it just needs to be impedance balanced.
Cable screen or not?
The screen causes interference to "dose" both wires more equally but, a bigger advantage is that it acts like a Faraday cage. This results in a lower common-mode interference voltage level and generally means that a differential amplifier will cope better in "harsher" circumstances.
Twisted-pair or not?
Twisting the wire pair is a satisfactory means to ensure that magnetic noise induction (interference) couples equally to both wires and, produces only a common-mode noise voltage signal at the inputs of the differential amplifier.
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