This question stems from my reading a great (and, as far as I can tell, thoroughly researched) historical novel about a Jewish family in Switzerland; ["Melnitz" by Charles Lewinsky] (http://www.worldcat.org/title/melnitz/oclc/904268308), now available in English as I just discovered - highly recommendable!
In it, one character converts to Protestant Christianity in an attempt to be better accepted in the notoriously anti-Semitic business community while telling himself he just finds the Christian practices more "reasonable"; one of the issues being the fact that Christian priesthood is more like a job in which one can be trained, as opposed to the hereditary priesthood of the Jewish Kohanim. (He thinks of his father-in-law, Mr. Kahn [!], who strikes him as utterly un-priestly…)
This is indeed a striking difference, so I was wondering whether there were ever - be it in late antiquity, the middle ages, the 19th century, or whenever - instances of, for example, Christians using the hereditary priesthood as an example of Jewish "backwardness", or of Jews pointing to Christians' abandonment of that idea as a sign for the wrongness of their religion etc.?
I've never seen Christians point this out, but that doesn't mean it didn't happen. What did happen is the priesthood being eliminated in Reform Judaism in the 19th century, which continues in that denomination.
But overall I think this makes less sense than you might assume it does, if that makes sense. In Judaism after the Second Temple, priests are just not important religious figures. The clergy are Rabbis, and priests are sorta secondary for a couple reasons. First, with the temple went pretty much all their rituals. Second, the only Jewish group to substantially survive the Second Temple era were the Pharisees, whose main religious authority figures were the Rabbis, not the priests. They've still got a ritual role, but it's fairly limited, and it doesn't give them any religious authority--maybe a bit of prestige, but it's just not a huge thing.
Really, setting up Jewish priests as equivalents of Christian priests is a coincidence of translation. People translate Kohen as "priest" because it's a handy term for a religious authority, even though there's really no connection and not much similarity between the roles. In Jewish languages the word for "Jewish hereditary priest" is "kohen", and there's some other word for "Christian priest" (komer in Hebrew, galekh in Yiddish, I've lost my paper on it in a computer change but I believe in Ladino it's papaz. If someone wants to help me find it, it's on terminology for other religions in Jewish languages, primarily Ladino, and is called "do not speak their names" or something).
So while people definitely have said that hereditary priesthood in Judaism is obsolete and have sought to change it, actually setting it up as a Jewish-Christian difference is not so common because they're really not equivalent roles.