The main reason for using twos complement to represent negative numbers is to make it easy to add (and subtract) two signed numbers. Adding 17 and -4 looks like this in binary (showing only 8 bits to keep it short):
0001 0001 (17) + 1111 1100 (-4)
= 0000 1101 (13)
It is more expensive to basic arithmetic without using twos complement.
The reason there is one more negative value to a signed integer type is because we use twos complement, not the other way around.