Mishnah: These are the things which separate Beis Shammai and Beis Hillel with [respect to the laws of] a meal.
Beis Shammai say: [In Qiddush] bless the day [i.e. make the berakhah referring to the qedushah of Shabbos], and then bless on the wine.
And Beis Hillel say: Bless on the wine, and then bless the day.

גמרא מה טעמהון דבית שמאי שהקדושת היום גרמה ליין שיבוא וכבר נתחייב בקדושת היום עד שלא בא היין מה טעמהון דבית הלל שהיין גורם לקדושת היום שתאמר ד”א היין תדיר וקדושה אינה תדירה

Talmud: What is Beis Shammai’s reason?
The sanctity of the day causes that the wine be brought, and one is already obligated to sanctify the day even when the wine hadn’t yet arrived.
What is Beis Hillel’s reason?
The wine causes that the sanctity of the day be declared.
Another thought: The wine is [relatively more] frequent, and the sanctity is not [as] frequent [and there is a rule with mitzvos, that all else being equal, the more frequent one is done first.

– Yerushalmi Shabbos 8:1, 56b

Rav Shelomo Yoseif Zevin (LeTorah uleMo’adim) comments on a famous dispute between Beis Hillel and Beis Shammai, and from it establishes a basic pattern to their philosophies. The most preferred way of lighting Chanukah lights is to differ the number with the number of days. We follow Beis Hillel, and light one on the first day, two on the second, etc… We “increase and proceed”. Beis Shammai, however, started with eight on the first day, second on the second… — “decrease and proceed”. Why?

Rav Zevin notes that there is a general pattern in their disputes. Beis Hillel see the situation in terms of what is, whereas Beis Shammai look at the potential. Since with each day, we added to what was, we also add lights in our menorah. Each day the miracle was greater than the day before. Beis Shammai look at the potential inherent in the remaining oil of the original Chanukah, as well as in our remaining Chanukah celebration. Each day there is less to look forward to, less opportunity before us. Therefore Beis Shammai reduce the number of lights as we progress.

A second, more common explanation of the difference in their basic philosophical orientation:

Qabbalistically, Beis Shammai is described as embodying the sephirah of Din, strict Justice and uncompromising Truth, whereas Beis Hillel draws from the sephirah of Chessed, Generosity and Lovingkindness. This fits the observation that Beis Hillel is far more often the more lenient of the two. Also, we are told that Beis Hillel’s position was codified as law over Beis Shammai’s because Beis Shammai would only teach their own position, whereas when a member of Beis Hillel taught, he started with Beis Shammai’s position, and then his own. Procedurally, we follow Beis Hillel because they were the larger school, and halakhah follows the majority. Beis Shammai was a smaller school that had stricter entrance requirements. Also, Chessed vs. Din. But their attitude might also explain how Beis Hillel grew more rapidly.

Furthermore, the reason given for the radical increase in the number of disputes between the generation of Hillel and Shammai and those of their schools is “shelo shimshu es rabosam — they did not properly serve their mentors.” They learned facts from their rabbeim, but without spending the time that comes from watching them live, they didn’t learn attitude. The Maharal explains that since Hillel was the nasi, his job was to distribute funds and build an infrastructure for society. His job was Chessed. Shammai, as the head of the court, had Din as a profession. The students, because of their distance from the rebbes, could not separate the differences due to their roles from the rebbeim’s approach to Torah.

Taking thes to our opening dispute, with Beis Hillel as explained by the first opinion in the gemara…

Beis Shammai say the order of blessings in the night-time Qiddush is the order in which the obligations arrived. First it became Shabbos, then you sat at the table with the cup of wine. Therefore, Qiddush should start with the sanctification of Shabbos and end with the blessing on the wine.

Beis Hillel instead focus on the order in which we are able to fulfill each obligation.

So Rav Zevin’s explanation works. Beis Shammai follows the order in which one gains the potential to do the mitzvah. Beis Hillel follows the order in which one can actualize that potential.

Also, the Sephirotic interpretation: Beis Shammai look to the obligation, the chiyuv — which also means “debt”, even though the person has no ability to fulfill it. So, blessing Shabbos comes first. Beis Hillel take a more generous approach, and don’t consider such a chiyuv to be fully real. Therefore, the sequence is when one can act upon it.