Tobold has asked recently why developers should bother mixing PvE and PvP into the same MMO, because he feels that the design of each conflicts with the other too much, leading to a reduced experience for both.
The simplest answer to this question is because people like to do both.
The next simplest is because making a whole new game would cost much more money. A game that includes PvE with no PvP is a massive waste of resources for a developer. Contrary to the opinions offered by some people, the vast majority of players enjoy PvP in some way - simply not PvP in an MMO. The negative connotations related to PvP in an MMO can be attributed to experiences of being "ganked", i.e. steam-rolled in a fight that you had no chance of winning or may not have even wanted at the time, because
a) the other player(s) was a higher level than you; and/or
b) you were outnumbered; and/or
c) the other player was able to attack you where/when you did not wish to be attacked.
For a), the solution in an MMO is obvious - make the achievement of maximum power something that is within the reach of all players, i.e., make any time investment required moderate (say, the same amount of time it would take in a single player RPG). This has the pleasant by-product of also fixing the retarded soul-sucking grind progression system that is PvE in many modern MMO's.
For an example of how to do this, as in many things, the answer is to look to Guild Wars. In Guild Wars it takes an easily achievable amount of time to reach maximum level. Most of the game happens at maximum level. Instead of arbitrarily increasing your power so that you end up fighting level 40 rats in between every town, you instead unlock new builds to try. Instead of forcing you to grind mobs/quests to obtain powerful enough gear to proceed through the game, you can easily achieve the most powerful gear and it's the good *looking* gear that you can grind for if you choose to. There is still all the e-peen stroking you can shake a stick at as a result, and all the other standard MMO stuff such as exploration, socializing, achievements (titles/emotes), super challenging areas with enhanced rewards (i.e. raids), etc.
Problem b) is another raised by Tobold. This applies to the 'open world' type of PvP, which often is and absolutely should be full of n00bs running around in big groups zerging each other. It is absolutely ok for the war as a whole to be asymmetric, dependent on who has put in the most time, etc, as this is the way that real wars are and let's face it, it's impossible to balance open world PvP any other way.
Two concessions must be made however. The first is that there must be more than 2 sides, as WAR's bleh PvP has shown.
The second is that there should always be tactical options available to players to counteract the movements of the other side. Games are no fun when no action that you could possibly take will effect the outcome. If your side has fewer players, it should be possible to accomplish various tasks to recruit more NPC's into the army (there should also be solid social networking tools, to organise the actual players that you do have more easily). If the other side chooses to zerg, tactics should be possible so that the enemy is forced to either
i) divide; or
ii) deal with the tactic somehow to avoid having to divide; or
iii) lose the battle/war
WAR has none of these factors and so can not possibly be a satisfying PvP experience (other than the thrill of running around with a large group, sieging a castle, etc, which I admit is quite fun the first few times). But these factors really don't get in the way at all of a solid PvE experience, so the good news is that future MMO's can easily adopt them.
Finally, problem c) above. There are many ways around this. The RvR 'lakes' in WAR were actually a good way to do this, and I hope future games adopt a similar solution.
Bear in mind that none of these problems are present in the more 'arena-based', e-sports style of PvP such as is found in Guild Wars. However for a long time, there were still balance issues where a skill that needed to be nerfed in PvP ruined popular use of PvE skills, or vice versa, making both camps generally unhappy whenever nerfs rolled around. This is another potential area of conflict between PvE and PvP game design.
However, again, Guild Wars eventually got around this - there are now simply skills that have an altered function when engaging in PvP. There are also skills that can only be used in PvE. This allows for fun gameplay in both areas, and for balance changes for one not to affect the other. Of course this has the down side of making the game more complex than necessary (and preventing people from smoothly transitioning to try out their less-preferred game type), but has the up-side of keeping a lot of people happy.
So in summary - PvE and PvP can and should be mixed into the same game, to provide a better experience for players and an efficient way to ensure game longevity and broader appeal. A little bit of common sense drawing from what's worked and what hasn't in previous titles is all that is necessary.