Interesting input from everybody until now.
I play as orthodox because I really don't like the Pope and all the game mechanics related to it - but other than this, the realm management is pretty much similar and people already explained above what you have to lose in terms of external politics. So for me the religious aspect is a bit of a plus, however, in catalan, occitan & italian regions you find interesting characters and scenarios worth playing - especially republics now, as soon as the DLC is released. Until this point, most people playing orthodox that I read about in the forum were either trying to keep the BE alive, actually restore its glory, or make a Russian empire much sooner in history. There are also some tough challenges as orthodox (ie. playing Georgia - surrounded by BE, muslims and the hordes.... ouch).
* What I didn't read above is the ability to reunite the churches and restore the Roman Empire, but I guess that is too localised on the BE, not orthodoxy itself.
* You can also pluck the eyes of your prisoners to teach them lessons.
* The orthodox have their own religious specifics, like the pentharchy; but they don't give the game too much authenticity, religion gets out of the way mostly - which I like.
* As the BE, you won't need the holy orders IMO, since the empire, if well managed, can produce enough troops and is easier to keep together in comparison to the HRE - IMO.
* If you lead many wars against anybody of any religion and brake your truce too often it will indeed NOT affect your game - which makes the game a bit more focused on staying alive against the hordes and muslims, historically a bigger problem than christians.
* One big advantage in 1066 start in the BE is that you are technologically more advanced than the catholics and you have some holdings which are already developed.
Playing a catholic or muslim has more flavour in my opinion if you like the events and all the micromanagement of the realm (I play Saladin as the king of Egypt right now), playing orthodox was more interesting to me when playing the warmongerer card. Actually muslim is the best in this regard since they can invade/holy war their way to world domination.
Disadvantages:
* the 1100-1400 period has deprived us, the posteriority, of a lot of information on the orthodox regions; there are many incongruences in the game in these regions (justified, since there's too much useless polemic about it and you can't sort it out objectively), so if you like to read the wikipedia about the surrounding regions and characters actual historical accounts, you might get confused by some mismatches.