The Mishnah in Shkalim (1:4) records a Machlokes regarding Kohanim:
א״ר יהודה העיד בן בוכרי ביבנה כל כהן ששוקל אינו חוטא. אמר לו ר׳ יוחנן לא כן אלא כל כהן שאינו שוקל חוטא.
R' Yehudah said, Ben Buchri testified in Yavneh that any Kohen who shuckles doesn't sin. R' Yochanan said to him, it is not so; rather, any Kohen who doesn't shuckle sins.
What is this machlokes? Why does Ben Buchri hold shuckling is allowed and R' Yochanan holds it's required? Why is the argument limited to Kohanim?
This is obvious. The question of whether or not one should shukl, is a well known dispute among the authorities. Some, such as the Sefer Hassidim hold it is a mitsvah that one must do, in a accordance with the verse "all my bones shall say, who is like you" while others, such as the Arukh HaShulhan and Mishnah Berurah hold that it is not necessary; rather, whatever is best for one's concentration should be done.
Ben Bukhri holds like the Arukh Hashulhan/Mishnah Berurah, while R. Yohanan holds like the Sefer Hassidim.
Ben Bukhri's point in stating that it is not a sin, is to disagree with the view of the Ba'al Shem Tov, et al. that shukling is inappropriate and disrespectful. Rather, asserts Ben Bukhri, it is permissible. (While R. Yohanan holds it is actually required).
That much is obvious, the question of why they speak about kohanim in particular is less clear. Perhaps Ben Bukhra's point is that even kohanim who are supposed to live on a higher level and act formal and proper as spiritual leaders, may nevertheless sway, an act which we have noted, many consider to be disrespectful.
As I see that the first question was answered, above, I'll explain why just Kohanim. The answer is simple, actually.
When Kohanim duchen, they have to hold their hands up and this causes the tallit that's covering them to weigh down on their fingers. Thus, there is a constant struggle that they have between keeping their hands up and the weight of the tallit. This causes the shuckling.
The makhloket is whether they need to hold their hands up or not, bedieved, if the weight of the tallit makes it difficult to do so.
