World of Warcraft has two distinctly different types of play: Player versus Environment and Player versus Player. The styles of play are dramatically different and there are few, if any, skills that cross over from one style to the other. WoW started out as a PvE game, adding in PvP content as the player base expanded. And despite the numerous PvP fanbois out there, the real game in WoW will always remain the PvE game.
There are a few reasons why I think this. First and foremost, you cannot progress in PvP without first completing a large potion of the PvE content. You start out at level 1 and progress up to level 70. You don't level up by PvPing against one another. You level up by fighting against the environment. Put simply, without the PvE there would be no PvP.
Secondly, PvP is an addition to the game. If you remove PvP from the game entirely, the game itself would not fundamentally change. However if you remove the PvE elements, the game would be nothing like it is. Everything would just exist like the Arena Tournament server. That might be fine for some people, and this is evident in the success of the Arena Tournament server. Even I enjoy spending a couple hours a week on there, but by no means would I want to just exist on a server where the only thing to do is kill one another.
Third, PvP progression is based on the amount of time one puts into it. The more time you put in, the better your gear is, and the better your gear is, the more matches you're going to win. The arena rating system somewhat offsets this by adding in a certain skill factor – but on the live realms there can be so much disparity between gear that some matches (and thus some arena ratings) become almost a joke. On the flip side, PvE is based mainly on the combined skill of 24 other people working together to take down a scripted encounter. These encounters often have random elements inserted into them (just as PvP is random) alongside the scripted ones. You can put in thousands of hours into the PvE game and not progress past Kael'Thas if you don't have enough skill; you can get a full set of good epic armor by putting merely a hundred hours into PvP. In no way can you do this in PvE. Of course, there are always those PvE players that magically float by and kill Illidan without doing any real work (I know a couple), but those are the exception and do not disprove the rule.
Finally, the gear necessary for PvP success does not translate to gear necessary for PvE success; however good PvE gear does mean good PvP gear. For example, you can have a full set of Season 3 PvP gear and the best PvP weapons in the game. If you walk into a Tier 6 raid you will fail miserably at what ever you attempt to do. You cannot tank any mobs well, you cannot heal well compared to PvE geared healers, and you cannot DPS nearly as well as those with Tier 6 gear. However, if you are completely outfitted with Tier 6 gear, you can easily walk into an arena or battleground and, for the most part, compete and have major success. My PvE Warrior DPS gear – which is a combination of Tier 5 and Tier 6 pieces – is enough that I am successful in both arenas and battlegrounds (especially in battlegrounds). The people that are able to kill me in a battleground match are few and far between. I simply do so much more damage faster and in larger crit amounts then my PvP counterparts that they just can't keep up.
While it is true that in individual instances there are pieces of gear and skills that PvP brings to the PvE table, the vast majority of the time the rewards for doing well in PvE content greatly outweigh the rewards for doing well in PvP content. You can't have PvP without having PvE first, and those PvE skills you acquire are always going to outweigh your PvP skills. This is the fundamental reason why the PvE game has been, is, and always will be the only real game.