Does being married by an Orthodox rabbi — as compared to being married by a rabbi of another denomination — afford certain privileges in today's world? My question pertains to the marriage of two Jews (not converts). Examples: Aliyah, synagogue membership permitted, the enrollment of future children in Jewish school/camp.
Documentation of marriage by an Orthodox rabbi will generally be useful to prove that you are Jewish and married, e.g. if you move to Israel. Required? No. But it can make the paperwork process easier.
Otherwise, our communal institutions tend not to have a hangup about "oh they're evil people and we must punish/shun them because they had a non-Orthodox marriage!"
Many Jewish dayschools will ask for proof that a child is Jewish before enrolling them; again, if you write on the application forms "we were married by Rabbi OrthodoxGuy", that should be all that's necessary, assuming your child isn't adopted or the like. (But if you can otherwise prove that your child's mother is Jewish, the school will enroll your child -- the average Jewish day school doesn't care if the parents are technically married according to Jewish law, only that the child is Jewish.)
Similarly, if you and your spouse can document your Jewish status, the average Orthodox Union synagogue will grant membership. (Again, documentation of an Orthodox wedding will generally suffice, but there are other ways.) If you join an Orthodox synagogue, at some point the rabbi may ask (hopefully nicely, and without any threats) you if you wouldn't mind spending fifteen minutes to do a nuts-and-bolts Orthodox wedding ceremony.