the way I learned it, it's just a notational difference. either way, you are dividing the divisor into each place of the dividend (starting with the most significant digit) and carrying over the divisor to the next place. short division is just a more implicit way of doing this that saves space. the shorthand version doesn't really work for divisors with more than one digit though.

What doesn't work with short division and multi-digit divisors?

it's not that the algorithm "breaks"; it's just that it stops being a convenient notation for humans.

the "short" notation relies on the ability to squeeze the remainder in between the digits of the dividend. if you have a divisor greater than ten, it's possible to have a remainder with more than one digit. two or more remainder digits is a lot harder to write legibly between the digits of the dividend. you also end up subtracting larger numbers in your head, which is error prone. finally, most people only memorize the multiplication table up to 12x12 or so. once they can't simply do a lookup into their memorized table, they switch to more of a "guess and check" approach, where they will inevitably have to cross out or erase their previous work.

in the end, it introduces a lot of unnecessary opportunities to make mistakes and hides stuff that people can't do reliably in their heads.

It’s notation only. I “invented” it as a kid to save space on division problems and reduce the chance of copying errors. It’s the same method as long division (and TIL it already existed!)

It definitely looks nice, and I'll probably start using it. If something is too inconvenient to use short division for (e.g. large divisors), I'll probably just use a calculator.