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 Lvush says the reason is because Avinu Malkeinu is parallel to the middle blessings, which we don't say on Shabbos. The Magen Avrahom says this reason doesn't explain not saying Avinu Malkeinu Friday afternoon though.
Because we are not allowed to ask for material needs on Shabbos. The Shaar Hakolel writes that according to the Siddur of R' Shneur Zalman of Liadai, a day in which we don't say Tachanun we don't mention sins or pray for physical needs, unless the prayer was instituted in the time of the Gemara.
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:
- The Lvush writes that Avinu Malkeinu is said after Shabbos is technically over.
- Even if one reaches Neilah during the day, because it is the time of the finalizing of the judgement we could rely on the Beis Yosef.
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