These amplifiers are designed to amplify a very small "sense voltage" of 10 to 100 mV, in the presence of very large common-mode voltages of 5 to 30 V. DC precision (low input offset voltage) and high common-mode rejection ratio (CMRR) are distinguishing characteristics of these amplifiers.
If a voltage outside the power supply rails is applied to the input, internal ESD protection diodes turn on, causing large currents to flow and damage these parts.
Specialised current-sense amplifiers, by contrast are designed so that, when powered from a low-voltage power rail such as VCC = 5 V and VEE = 0 V, they can withstand pin voltages much higher than VCC and much lower than VEE.
These amplifiers use specialized ESD structures that enable them to have this functionality.
In special cases, no VCC pin is required to accomplish current-sensing; the MAX9938 is such a device.