Our Gemara on Amud Beis brings the verse that directs us to treat cohanim with extra honor, and in fact directs the Beis Din to enforce the priestly marriage restrictions. The Mei HaShiloach Parashas Emor discusses the nature of the distinction of the Cohanim, and even the nature of the distinction of Jews in general:

His first point is based on the Gemara’s interpretation (Chulin 57a) of a verse in Zecharia (10:4). Though Cohanim are obviously of higher status in practice than a Yisrael or Levi, their status stems from the holiness of the Jewish people as a whole. He notes a dispute in Mishna Middos (2:6) about whether the section in the Temple for the Cohanim is elevated or not and makes it dependent on whether the holiness of the Cohanim is intrinsic above the regular Israelite. The position of the Tannah Kammah of that Mishna has the sections on the same level of elevation, symbolizing that Cohen and Israelite are on the same plane.

Then, Mei HaShiloach raises an intriguing question.  Every other group in Jewish society gets its own daily blessing of thanks, such as Shelo asani Eved, Shelo Asani Goy, “That you did not make me a gentile or a slave.” If so, why is there no blessing, She-asani Cohen? To answer this he makes another interesting observation regarding these daily blessings.  The language is not, She-asani, but shelo-asani, that is, “He has not made me….”, as opposed to “He has made me…”. The reason for this is that ultimately, who we really are and our holiness is a matter of our choice, and we must earn it through our actions. That is why, we should not say, “She-asani”, that Hashem made us…because we have to make ourselves. We can only thank him for positioning us and NOT making us something of a lesser quality.  This explains why a cohen does not recite this blessing. He cannot recite, “she-asani Cohen”, because that is not the structure of these blessings, for the reason stated. But he cannot either bless, “shelo asani Yisrael”, because quite clearly, a cohen also has kedushas Yisrael.