The difference between the derivative is the same as the difference between the slope of a line and its eqn.

The derivative is the slope

Thelinear approx is the the tangent line itself

derivative -- f '(a)

linear approximation y = f ' (a) (x-a) +f(a)

The derivative is the instaneous rate of change so whenever you want the inst rate of change of a function.

an example of where the linear approximation is used in error analysis once you learn the differential, which is the change in f(x) as you move along the tangent line.

If you want see the notes on the differential and local linearity

at Calculus 1