What are actual additional benefits of using high voltage
If you are referring to power transmission, then you have put your finger on exactly why Tesla beat out Edison back in the day.

When George Westinghouse was called upon to provide power to New York for industry he decided that Niagara Falls was the best way to get the power.  His original idea was to pipe compressed air.

Edison had his electricity plants but they ran on DC, which means the power must be generated at the same voltage at which it was used.  To keep the number of home electrocutions to a manageable level this meant that it had to be kept in the 100-200 volt range.  With available cables the power could only be transmitted a few miles before the voltage drop in the lines made it impractical.

Remember:  Power loss in cable is defined by the square of the amperage multiplied by the resistance.  

$$ Ploss = I^2 \times R $$

where Resistance is determined by the cross-sectional area of the cable. 

And since power is the product of the voltage and amperage, 

$$ P = E \times I $$

Edison's power-delivery system was current-limited due to cable losses.

So then along comes Nikola Tesla with the idea of using what he called "rotating magnetic fields" - we now call it triple-phase power.  By using this he could use AC in transformers to step up the voltage to very high levels.

With his system he could deliver more power at the same amperage via the same size cable by using high voltage.  

This translated directly into the ability to install long-distance transmission lines.

Westinghouse immediately saw the advantage of this, so thus Niagra Falls became the very first electrical power plant to transmit power for such a long range.  

At the point of use it could then be stepped back down by transformer to a level that could be used in the premises.  

Some very long-haul lines can be up over 100,000 Volts.  At substations it is brought down to lower levels for local power lines.  

Then the transformers on the poles bring it down to the 240 Volts that feed our homes.

On our end, as consumers, we see that by using AC and its ability to be stepped up to high voltage has given us the benefits of the modern power grid system.  

Thus we now have AC outlets in our homes as a result - with power that is being provided from far away.