The solution Catalin posted works with the problem you described in your first post (nothing more than a count of days), within the limitations I indicated in my first post.
Your second post relates to an entirely different scenario. You can't expect to get the 'right' answer if you don't provide all the relevant detail. As for DATEDIF, your assertion that it "
does nothing different or new compared to simple substraction of the cells" shows you really haven't taken the time to learn how in works - it does far more than a simple addition/subtraction can do. See:
http://www.cpearson.com/excel/datedif.aspx
Given the additional information you've now provided, which shows that you're actually trying to calculate an age based on the difference between two dates (in A1 & B1), try:
=DATEDIF(A1,B1,"Y")&" years, "&DATEDIF(A1,B1,"YM")&" months & "&DATEDIF(A1,B1,"MD")&" days"