Battlegrounds focus on how well you can work as a large team, thus your individual choices aren't as important as say an Arena, where as an individual player, what you do matters a lot more to the outcome of the match.
Battlegrounds: Focus on what a large group of players can do in synergy to win, there's more room for error.
Arenas: Individual oriented fights where there's less room for error.
Skill in the battleground sense is defined by objectivity and teamwork, you don't have to be BiS or be very skilled to do well if you play objectively and with the team.
Skill in the arena sense is about actual player VS player and mastering your class to it's best ability, both gearwise, gameplay wise, and minmaxing your damage, burst, and survival.
From a teamwork perspective? Battlegrounds require more skill. One player isn't going to win the game.
From an actual PvP perspective? Arenas, your choice as one player matters a lot more.