Amplifiers serve many functions; making a signal larger, or smaller, or
changing from low impedance to high impedance, or from high impedance
(current source) to low impedance (voltage source).
One useful function of a buffer (voltage gain of 1) amplifier is to prevent interfering signals
at stage 2 from propogating to stage 1 of a signal-processing chain.
So, a buffer between a source and a long wire connection will serve to
prevent antenna-like pickup on the wire from disturbing the source.
Buffer outputs can be safely probed without disturbing
device operation.
A buffer output can drive a shield to minimize or null stray capacitive
currents.
One might use a buffer before an element that has multiple
states (like, low-power sleep mode) to protect the signal from being
corrupted by the dead element, or during power transitions, or to prevent the signal from powering-up circuitry intended to be inert.
Another useful function is to limit the signal; a buffer can deliver output
that is guaranteed to be within a known range (voltage, current, slew rate limits)
for input to low-speed, low-voltage, or logic elements that are intolerant of some
potential signal characteristics.
Finally, integrating power output onto an operational amplifier chip
runs a risk of thermal feedback effects; a buffer amplifier with
heatsinking can be the perfect output stage, thermally distant from the
sensitive input pins of the first-stage amplifier.