First and foremost: At the end of the season, Frostmourne characters will be transferred to Icecrown, so it's all the same in the long run.
Now that we've gotten that out of the way,
Frostmourne:
+ Lower-gear brackets for battlegrounds and arena (iirc)
+ Less gear disparity between you and the rest of the players, less inflated GS (see the Icecrown section)
+ More challenging bosses (better to help you learn the ropes because you can only get carried so much)
+ No queue
+ Progression, 2 raid locks a week!
- More challenging bosses (can result in a steeper learning curve)
- Will soon enter ToC patch (kinda 'meh' raid compared to Icecrown Citadel)
- Progression also means more responsibility, since you need to keep up with your peers to progress the current raid tier (especially if in a guild). And 2 raid locks = more time spent raiding per week.
Icecrown:
+ You have immediate access to endgame raids as well as catchup mechanics to help you gear up (this means that by the time Frostmourne characters are transferred, your Icecrown toon will likely be much higher geared).
+ Base difficulty bosses + high gear allies = more carrying, so less responsibility on your side
+ Static endgame state = No time limit, you can do whatever you want at your own pace
- Inflated raid requirements: Many PuG raids' GS requirements can be very difficult to reach without having gear from that exact raid, and without achievements you probably won't be given entry to a raid either. Entering a guild mostly solves this problem, but without one, you'd likely find the raid requirement wall to be very frustrating.
- Base difficulty bosses + high gear allies = more carrying, so you don't learn as much
- Many Best-in-slot characters: You'll be playing the catchup game for a long, long time
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Imo it depends on what you look for; if you're looking for more progression raiding and have the time to put into learning everything and all then Frostmourne will probably serve you better, but if you just want to chill or have more limited time then Icecrown is the choice I would recommend.
PvP, I think Icecrown is a bit *too* saturated with endgame geared bois, so Frostmourne might feel better to start in (then again, you can also Blackrock if PvP is what you want).