Why do we not say Avinu Malkienu on Shabbos and why is the Teffilah of Neilah the exception to this very rule?
The Magen Avraham gives two reasons:
The Shulchan Aruch rules that we do say Avinu Malkeinu on Yom Kippur that falls out on Shabbos. The Rama says that our custom is that we do not.
The Magen Avraham says that by Neilah there are two reasons why we say Avinu Malkeinu even when Yom Kippur falls out on Shabbos:
We don't say it at most tfillot because we avoid making requests on shabbat.
As for the second part: Maybe because Ne'ila as a tfilla is an expression of our extreme desire for kapara that the avinu malkein then is like saying a mishabeirach for a really sick person on shabbat -- ie we do it anyway with 'shabbat hi milizok' OR Maybe since we always get to that part after shkiya we are more meikil