We continue with the Mishnah in Berachos, which presents another dispute between Beis Shammai and Beis Hillel regarding the meal: which comes first, washing the hands or pouring the cup?
"Beis Shammai omrim: notlin la'yadayim v'achar kach mozgin es hakos" - first one performs netilas yadayim, and only afterward pours the cup.
"U'Beis Hillel omrim: mozgin es hakos v'achar kach notlin la'yadayim" - the reverse: first one pours the cup, and afterward washes the hands.
What is the Mishnah discussing?
For the sake of simplicity, we will follow the Rif, who explains that the discussion concerns the wine of Kiddush that precedes the meal on Shabbos and Yom Tov. The question, then, is which comes first. This very distinction exists in practice today as well: some wash their hands for HaMotzi and then make Kiddush, while others make Kiddush first and only afterward wash their hands for HaMotzi.
The root of the dispute in the Gemara:
The Gemara explains that the basis of this dispute lies in another question about which Beis Shammai and Beis Hillel disagreed: using a cup whose outer surface is tamei.
Beis Shammai: it is forbidden to use a cup whose outer surface is tamei.
Beis Hillel: it is permitted to use a cup whose outer surface is tamei.
The approach of Beis Shammai:
Beis Shammai were concerned that liquid from inside the cup would spill onto its outer surface. As long as a person has not washed his hands, his hands are presumed to be tamei with the status of sheni l'tumah, and a sheni l'tumah that touches a liquid renders it a rishon l'tumah by rabbinic law. The halachah is that a liquid which became tamei by rabbinic law has the power to render the outer surface of a vessel tamei. And since this is only a rabbinic tumah, the Sages said that only the outer surface of the vessel becomes tamei, but its inside, its rim and its handles do not - only the outer part is considered tamei. And since in the view of Beis Shammai it is forbidden to use a vessel whose outer surface is tamei, they were concerned about this mishap, which could occur as long as the hands have not been washed.
And why do Beis Shammai not permit using a vessel whose outer surface is tamei? Because of the concern that after a person has washed and his hands are tahor, liquids will spill onto the outer surface of the vessel, his tahor hands will touch those liquids, and those liquids - which became tamei through contact with the outer surface of the vessel - will render his hands tamei. Therefore Beis Shammai say: wash your hands first, in order to ensure that the outer surface of the vessel remains tahor.
The approach of Beis Hillel:
Beis Hillel, on the other hand, hold that it is permitted to use a vessel whose exterior is tamei, and therefore they were concerned about a different and greater problem: if a person washes his hands first, perhaps his hands will not be completely dry of the washing water. And since in their view it is permitted to use a tamei cup, the liquid on the hands could become tamei from the exterior of the cup and render the hands tamei - so that even after washing, he would eat with hands considered tamei, whereas the entire purpose of washing the hands before the meal is that the hands should be tahor.
An additional reason for Beis Hillel's view in the Beraisa:
The Beraisa offers an additional reason for the view of Beis Hillel: "Teichef l'netilas yadayim seudah" - one should not interrupt between the washing and the meal, just as one does not speak between netilas yadayim and the berachah of HaMotzi. There is value in the washing of the hands being immediately adjacent to the berachah of HaMotzi, and therefore one makes Kiddush first and only afterward washes the hands for HaMotzi.
In summary: Beis Shammai and Beis Hillel disagree about the order of precedence between washing the hands and pouring the cup, and the root of their dispute is the question of whether it is permitted to use a cup whose outer surface is tamei. Beis Shammai, who forbid it, place the washing first so that the outer surface of the vessel will remain tahor; Beis Hillel, who permit it, place the pouring of the cup first so that wet hands will not become tamei from the exterior of the cup, and also because of the need for the washing to be immediately adjacent to the meal.