The primary advantage of two's complement over one's complement is that two's complement only has one value for zero. One's complement has a "positive" zero and a "negative" zero.

Next, to add numbers using one's complement you have to first do binary addition, then add in an end-around carry value.

Two's complement has only one value for zero, and doesn't require carry values.

You also asked how the range of values stored are affected. Consider an eight-bit integer value, the following are your minimum and maximum values:

Notation     Min   Max
==========  ====  ====
Unsigned:      0   255
One's Comp: -127  +127
Two's Comp: -128  +127