Why is the input impedance of an emitter follower defined as \$\Delta V_{B}/\Delta I_{B}\$, as opposed to \$V_{B}/I_{B}\$?
A more basic answer is that the resistance of the transistor is non-linear, and can vary significantly. For resistors, in general the resistance doesn't change over voltage. More specifically, the slope of the resistance doesn't change. Therefore V/I is dV/dI. 

For a transistor/diode, there a dynamic resistance that has a highly variable slope. A diode with a forward voltage of 100mV might have a resistance of 1MΩ, while at 1V it might have a resistance of 10Ω (random numbers chosen, but the general magnitudes should be relatively correct). The problem now is that resistance is very non-linear, which we don't like for many reasons. Thus, a small signal model is used, where the input voltage is changed a small amount, and the current is measured at 2 points. This change in current for a given change in voltage (slope) is the roughly the of the resistance at that biasing condition. 



In essence, resistors have constant slopes for voltage versus current, and voltage over current (slope) is equal to resistance. Transistors have a changing slope, so the resistance must be approximated at a given point with dV/dI.