The reason why we use two's complement rather than one's complement is to make arithmetic as simple as possible.
Consider that in one's complement, 1111 1111 and 0000 0000 are the same number - by subtracting one, we have ended up with...the same number. This is too big of a pain to have to think about - so instead we use two's complement, where 1111 1111 is -1 - by subtracting one, we successfully subtract one. Hooray!
(A secondary advantage is that we can represent one more unique number in two's complement than in one's complement — in two's complement -128 to +127 instead of one's complement -127 to +127.)
