A linkshell is a casual place to chat with friends and be social. A free company is more for progression, and is associated with grouping for events. This is also why free companies will be able to have their own housing, as it is more like a guild setting. It is a bit confusing having both in-game, but it's also kind of nice because it allows players to still be social in multiple linkshells, but really focus their progression oriented goals in their free company (which you can only have one of).