You are able to go out and quest at any point with your PvE (Player vs. environment) characters. Most of the level appropriate missions and quests are geared to full groups of 4-8 depending on how far along you are. If you can't find others to join you, or just want to play alone, there are computer controlled henchmen (tank, healer, archer, ...) that you can bring along to fill out the team.
As for the PvP (competition arenas), you can play through the entire PvE world without going into the arenas at all. In fact, it is possible to just fight through the explorable areas without doing any of the main story line missions. Or, you can even create pre-made Lvl 20 PvP characters using skills you have unlocked in PvE or by spending faction points to go to any of the arenas. These prebuilt characters are not allowed into the explorable world or missions, though.
The Lvl 20 cap is intended to make players focus on skill and strategy rather than grinding up to get to the next level all the time. Once you do get to 20, you do continue to earn skill points to learn new skills (special attacks or spells). The level cap also helps balance the PvP battles as well. I'm sure most of the people in the WoW battlegrounds are at or near Lvl60.
GW also limits players to 8 skills when in a mission, quest area or arena. This forces some level of strategy for selection of the skills.
I would say if you enjoy exploring new areas, WoW has a much larger world for you to explore. GW still has a lot to do, but is tuned a little more for the PvP. Don't get me wrong, though, I have been playing casually for several months and haven't completed the missions yet.
Wow, I got a little long winded there