What happens is the that the HI input actually cuts the input
signal, while the LOW input will leave the signal exactly as it is when
it enters the amp. But the reasons for choosing one or the other vary
depending on your instrument and your amp.
For example, some bass amps advise players using a bass with active
pick-ups to use the HI input, while it is recommended that those using
passive pickups plug into the LOW input.
As for guitars, the choice is also dependent on what type of pick-up
is used. Guitarists using new, bright humbuckers might want to choose
the HI input to tame the hot signal, while a player using vintage
single coils may want to go with the LOW input to ensure that all of
the true sound of the pick-up is replicated.
But, as is usually the case, there is no right answer, other than
the option that sounds better to your ears. The amount the signal is
cut in HI inputs varies widely from amp to amp (anywhere from 6 dB to
15 dB is common), and so do pick-up types. Test out which works best
for your instrument – some players use one jack for one guitar and the
other for a different axe (though it should be noted that for some
amps, including some Fenders, both inputs become identical if both
inputs are occupied).