What is the difference between differential resistance and resistance?
It's also referred to as dynamic resistance.

What you call the differential resistance is of importance when talking about a non-linear resistive element such as a diode as in the trusty 1N4148: -



Or maybe in the breakdown voltage of a zener diode: -



So, taking the 1N4148 example, you can see that I/V (or V/I) when the current is (say) 100 mA is significantly different to the slope at a current of 1 mA.


At 100 mA, the apparent resistance is 1 volt ÷ 100 mA  = 10 ohms
At 100 mA, the dynamic resistance is about 0.2 volts ÷ 10 mA = ~20 ohms
At 1 mA, the apparent resistance is 0.6 volts ÷ 1 mA  = 600 ohms
At 1 mA, the dynamic resistance is about 0.2 volts ÷ 100 uA = ~2 kohms


As you can see there is a big difference between apparent and dynamic resistance.

For a zener diode, as it starts to conduct, the slope of V/I changes quite a bit as it goes from non-conducting to "fully on". But it's the same basic story of apparent vs dynamic resistance.


  Also in this literature I'm reading, the authors choose to
  perform/read \$r_{diff}−I\$ plots (differential resistance vs current bias)
  for a current biased circuit


That's because the dynamic resistance varies with applied current (as per the 1N4148 diode).


  so why do they take the first one? And also what does it tell us that
  I-V characteristic doesn't?


Well, the full I-V characteristic does contain the information but sometimes it's helpful if someone has told you what the dynamic resistance is at certain current levels because it saves you working this out.