The higher the index, the thinner (and therefore the lighter) the lens. This
can improve the aesthetics of your lens if it is very high powered, but for
low power lenses it may make little difference in thickness or weight.
Lighter lenses can also improve how well your frame sits on your face (less
slippage down your nose), but again this has a bigger impact if the Rx is
relatively high in power.

The problem with high index lenses is that the higher the power, the worse
the optics. This is indicated by "abbe value" of the material which is a
measure of chromatic aberration. The higher the abbe value, the less
chromatic aberration. Ironically, chromatic aberration is more of a
perceived problem with a high power Rx, which is more likely to need a
thinner and lighter lens.

The one exception to the above is polycarbonate lens material (1.59 index),
which has the worst chromatic aberration (lowest abbe value) of any widely
used material, including those materials with a much higher index. Polycarb
does have the advantage of having high impact resistance and high tensile
strength, but Trivex (1.53 index) has the same general safety
characteristics, but has much better optics. Trivex material is somewhat new
and is available from Hoya, Younger, and maybe one other manufacturer.

The term "high index" is a bit vague and means different things to different
people. Find out the exact material and index that is being recommended to
you. If you want some advice on which material to get, post your exact Rx
here.