The difference between the two dates can not be represented (correctly) in the form of YY.MM.DD. It can be considered as suggested by Alexey Shimansky (difference of years; difference of months, if negative, decreasing by one difference of years; then days), but the result will be meaningless.
The problem is as follows:
1) Take 4 dates: a) 31-01-2015, b) 01-03-16, c) 31-01-2016, d) 01-03-2016. If you subtract the dates (bs and gs) as you propose, you will get the same result (0 years, 1 month, 1 day). But actually between dates in the first case 29 days, in the second - 30.
2) An example from my comment: 2015.03.01 - 2015.01.31 and 2015.03.31 - 2015.03.02 . The difference here and there is the same - 29 days. And you get different results.
Generally, subtracting one date from another, we lose the "binding" to a particular month and year. The result of subtracting two dates is simply a time interval that is not tied to a particular year. It can be expressed in seconds, minutes, days, but not in months or years.
The sum of two dates is completely devoid of any meaning, although it can be quite correctly calculated by adding the time since epoche and translated back into the date.