One advantage of the short scale is that its cadence is the same as that of SI units: million is mega, billion is giga, trillion is tera, and so on. The long scale of course has the -ard names, so this isn't exactly a problem, any more than French saying "four twenties and nineteen" for ninety-nine is a problem.
The difference is one of the major reasons the SI scale names exist in the first place, in fact.
It also matches the decimal separators used by everyone but Indians, which strikes me as more logical.
What's the advantage of the long scale? Just historical?
But really, each is just a quirk of languages. I don't expect 7 to sound like "seven" in other languages, why expect billion to sound like "billion" rather than "milliard" or "trillion"? When conveying information across languages, we use numerals, sometimes with the SI scales, which are universal.
Just another confusing faux amis to figure out when learning another language, really.
I was very surprised to learn about this the first time. In USA English:
10^3 -> thousand
10^6 -> million
10^9 -> billion
10^12 -> trillion
10^15 -> quadrillion
10^18 -> quintillion
But there's another scale too! See this Wiki article for more
https://en.wikipedia.org/wiki/Names_of_large_numbers