I've played both games for a couple hundred hours each. This is a long post, but I'll try to make it very comprehensive. Here's what I think of them:
If you want, scroll down the conclusion, which is in bold.
The strength of GW lies in its core gameplay. Combat mechanics and character development are done very well. There's a great deal of strategy and tactics, emphasizing skill over patience (players who grind for 100 hours aren't significantly more powerful than someone who doesn't do any farming). However, the world isn't as rich, the UI is less customizable, and there isn't that much content. There isn't much item variety and the economy is pretty simple.
WoW has an amazing, immersive world. The UI, the quests, the economy, and many other things are done very well. However, my characters never really had a very interesting selection of abilities and combat is pretty formulaic. However, the Auction House, crafting, instances, and other elements makes the game a lot of fun.
They're both excellent games, but they cater to very different play styles. WoW encourages grinding (spending several hours doing a fairly repititious activity, in order to obtain experience, items, or abilities) and tedious activities. Expect to spend a lot of time traveling. GW is the opposite. Even if you wanted to do something tedious, there isn't that much to do. Farming doesn't reward you with that much. You can instantly travel to any town that you've already visited, so you don't have to spend cash and time siting on the back of a Gryphon.
WoW started out as a great game. It is very fun for the beginning player. However, once I got to the midgame (about lvl30), it just got boring. I was spending more and more time doing the same things (Immolate, Curse of Agony, Corruption, Drain Life, Drain Life, Shadow Bolt, Immolate, Drain Life, Drain Life, repeat). GW is the opposite. My first couple hours were a lot of fun. However, up until about the 30th or 40th hour in, the game was good, but nothing stood out. It just didn't have the captivating setting or extracurricular activies that WoW had. However, once I reached the Desert, things became very interesting. The 8 skill limit, which started out as being annoying, became a mechanic that encouraged me to develop unique strategies. Enemies starting to require more attention and quick thinking. The very high end PvE areas force you to adapt on-the-fly, depending on what your team is doing. For example, Aatxes (a mob in the UW) can kill a tank in 3 hits and a caster in 2. They attack about once a second, maybe a bit less. When you're fighting 3 of them at a time, you really do have to think, in order to succeed. But, you know what? It's very possible. I'm not a good, but not great player, and I have had a lot of fun playing in the high end areas.
In WoW, I never got to experience the high end areas. It just takes a lot of time to reach time. On the one hand, this speaks for the vast amount of content that WoW has. On the other, it requires you to have a lot of patience.
PvP is very different in both games. WoW's PvP works in battles from 1v1, to 45v45, or even more. However, you succeed or fail based upon your class and your items, not really on your skill. A player with trinkets that protect against Charm or Fear, or an Undead Rogue, will absolutely decimate a Warlock. Paladins have a huge advantage against Warriors. Priests usually beat Mages. Druids beat Rogues, unless the Rogues have a Spider Belt, which gives them an advantage. It felt too much like the battle was over before it started. However, large scale battles were more balanced because of the vast number of players. Defending the Redridge Mountains can feel epic, at times. GW is opposite, again, in the PvP system. You only have the option of 4v4, 8v8, 8v8v8, or 8v8v8v8v8v8. There's no 1v1 or 2v2. There are no random encounters with enemy players in the countryside (that also means that, unlike in WoW's PvP servers, a lvl60 Orc won't happen upon your lvl30 Human and destroy you). However, it is supremely well balanced. You win or fail based around the strategies and tactics that you and your team execute. You win based upon your skill. It feels like it's actually you that's playing, rather than your items or your character. It's pretty easy to acquire the abilities and equipment in order to compete at a high level of play, but it is still takes intelligence, organization, and quick reaction times to master.
WoW and GW have different types of servers. In WoW, there are 50 different servers. If you make a character on one server, you won't encounter anyone on another server, with that character, ever. This means that you may end up on a server with 4 Alliance for every Horde (or vice versa). However, it also means that there's only one copy of the game. If Orgrimmar is under attack by the Alliance, you can't just switch to another district and avoid combat. You either have to retreat to another town or fight. You also can happen upon players while you're adventuring and communicate or help them out. It builds a greater sense of community. However, it also means that you could be ambushed and outnumbered by enemies who delight in killing easy targets (this is only on PvP servers; on the other servers, PvP is purely consensual). This system does make WoW prone to lag. GW has only three servers (Europe, America, and Korea). However, everything is instanced. This means that only a certain number of people can occupy a given area at any one time. Thus, there can be 50 different copies (districts) of Ascalon City. On the one hand, this makes it easy to find friends or guildmates ("meet me in district 27"). On the other, GW has less of a sense of community, because of it. While you're out in the field, you will never encounter other players, since a unique instance of the game is reserved for your party. This cuts down on the lag. You also never have to worry about another player killing the object of your quest, forcing you to wait until he respawns. Since nothing respawns in GW, you can take it slow, or safely wait in a previously cleared area for someone to go to the bathroom or grab a drink. It's much less annoying.
It's pretty easy to solo in WoW. Everything except elaborate, time consuming, optional dungeons, can be done by yourself. However, you rarely encounter more than one or two mobs at any given time and it is almost impossible to kill anything more than 4 levels higher than you.
It's very difficult to solo in GW. You have to build your character specifically for it. Even then, certain types of mobs will destroy you. However, free, AI-controlled henchmen are available in every single town in the game, so you will have no problems filling out your party if you can't find any real players or just don't want to play with real players.
In WoW, there's no real direction to the story. You aren't special "chosen ones" who follow a linear path. You're just another adventure, making a name for himself. Because of this, everything is optional. No quests are required, although certain ones will nudge you in the right direction, so it's hard to feel lost. In GW, it's the opposite. There is a strong (but crappy, in my opinion) plot that carries you through the game. In order to progress, you have to complete Missions that require anywhere from 30m to 2h, to complete. Still they're a lot of fun and have some interesting mechanics that you don't find in WoW's equivalent (instance dungeons).
Character creation in GW is superior to that in WoW, in my opinion. WoW offers more options, like crafting, a larger selection of items, and races. However, it's difficult to make your character feel unique. Every single Priest in the game plays similiarily to every other Priest. Every Mage feels the same. There isn't that much to differentiate how you act. GW is different, yet again. There isn't any crafting, or extraneous mechanics, and the selection of items is mediocre. Still, it does feel like you are making a unique character. There is a tremendous amount of strategy in making a build, yet it's a simple and rewarding process. Every character customization choice makes a difference in how the character plays (except for cosmetic issues, like hair color). A Minion Necro will play differently than a Blood Support/Damage Necro or a Debuff Curse Necro. A Knockdown Hammer Warrior/Elementalist will do great damage while hindering the enemy's ability to do anything. A Sword Warrior/Monk will be able to survive tremendous amounts of damage. The greatest part about the game is that everything can be easily changed, except for your name, physical appearance, and primary class. If you feel like your Sword Warrior/Monk isn't doing enough damage, then you can switch to a Knockdown Hammer Warrior/Elementalist, although you may have to search a bit to find a good Hammer and do a few quests in order to get some Elementalist skills. WoW is the opposite. You have to pay a LOT in order to switch around your build. Most things take a great deal of expense or effort to change.
GW has better community-developer relations than WoW. ArenaNet releases patches more often and fixes problems earlier, although it is about equally communicative (Blizzard employees post a lot more frequently in forums, but ArenaNet grants an interview to a different fansite every week).
So, if you're a combat-oriented person who likes lots of strategy, tactics, and skill, and can stomach a lack of out-of-combat niceties, take GW. If you like all the extra features that WoW has, like mail, crafting, an Auction House, an immersive world, and all that, and you have the patience for somewhat rote combat and a bit of tediousness, take WoW.
Did I mention that GW has no monthly fee?