What are advantages of Two's Complement?

Two's compliment representation of signed integers is easy to manipulate in hardware. For example, negation (i.e. x = -x) can be performed simply by flipping all the bits in the number and adding one. Performing the same operation in raw binary (e.g. with a sign bit) usually involves a lot more work, because you must treat certain bits in the stream as special. Same goes for addition - the add operation for negative numbers is identical to the add operation for positive numbers, so no additional logic (no pun intended) is required to handle the negative case.

While this doesn't mean it's easier from your perspective, as a consumer of this data, it does lessen the design effort and complexity of the device, thus presumably making it cheaper.

The ADC can convert data (say input voltages between 0 and 5V) and you either need that data to be unsigned (0V=0, 5V=max code) or signed (2.5V=0, 0V=max -ve, 5V=max +ve).

In addition to 2's complement being the commonest computer representation for signed data, the conversion between the two formats described above is completely trivial : simply invert the MSB!

This is incredibly cheap to add to the ADC's internal logic and gives the ADC another selling point on the datasheet...

If you need to perform math on the representations of negative numbers, twos complement makes that easier than offset binary, which will match with the "signed int" data type. Your compiler will simply know how to deal with it. Otherwise, you spend clock ticks converting back and forth.