Answer:
   
   A rabbi has no actual powers in the written Torah, although the Talmud
   does provide the Rabbi with the authority to make interpretations of
   Torah (which, in Orthodoxy, provides authority). Rabbis are, however,
   ordained (a term used in the progressive communities) or given
   semichah. This is a recognition of a certain level of training or
   education as defined as appropriate for the community in which the
   Rabbi has studied.
   
   One of the traditional names for semichah is hatarat hora'ah, which
   translates as a license to instruct. In the Orthodox community,
   semichah is granted in two forms: Yoreh Yoreh (to instruct) and Yadin
   Yadin (a higher level, meaning to judge). This was seen in earlier
   times. For example there was the "Magid" or preacher (the role of
   teaching Jewish law and judging being separated from moral
   instruction).
   
   Because of the rabbi's training, the rabbi often takes on other roles.
   Rabbinical presence at religious services is desired insofar as
   everyone likes the rabbi and the rabbi can rule on questions that come
   up related to the service (e.g. does a particular smudge render a
   Torah scroll unkosher?) If the rabbi has a nice voice, and no one else
   has priority, the rabbi may even lead the services. The state gives
   rabbis the permission to perform weddings and so on since the state
   trusts them.
   
   Priests are male descendants from Aaron, the brother of Moses. They
   are usually called cohanim [cohen singular]. The cohanim perform
   Birkat Cohanim (blessing the congregation using the Hebrew text found
   in Bamidbar [Numbers] 6:23-25) on the following occasions:
   Daily
          ...in Israel (except the Galil, per Minhag Tzefat)
   Shabbat and Yom Tov
          ...in many non-Israeli Sephardic congregations
   Yom Tov 
          ...otherwise (non-Israeli Ashkenazic congregations)
          
   Cohanim are traditionally granted priority in numerous details. They
   are also traditionally forbidden to attend funerals other than their
   closest relatives and may not marry divorcees or converts. When the
   Temple is standing, the cohanim run most of the Temple service.
   
   The "Star Trek" Vulcan "live long and prosper" sign is roughly
   one-half of the gesture the cohanim make when blessing the
   congregation.1 You can see it engraved on many cohen tombstones:
\\//_ _\\//
 \ /   \ /

   The Pharisee/Sadduccee conflict was a sectarian division in the period
   of the Second Temple, although some view it as a rabbi/priest
   conflict. When the Second Temple was destroyed, the priests lost most
   of their power.
   
   Oh wait, you meant maybe, like Catholic/Anglican priests? Heh.
   
   On this note: Priests are often used as intermediaries between man and
   G-d. Rabbis are nothing more than regular people who have learned much
   Torah. Catholic priests can give absolution for sins, rabbis can't
   (unless you're asking forgiveness for something you've done against
   the rabbi personally).
   
   On the other hand, in the traditions of the Chassids and in the
   Sephardi communities, holy men sometimes have a role as intermediary
   (though not obligatory, of course). The tales of Rabbi Levi Yitzchak
   of Berdichev are filled with stories of his intercession On-High. This
   was a dominant theme in Chasidic "maasehs."
   
   Footnote:
   
   1: The Vulcan's learned of this symbol from Leonard Nimoy, who is
   Jewish.