You can explore off the mission area.
The game has several missions tied to each world, currently only the Earth is active (with the moon having one mission open last night).
However, the worlds are basically one map. So you are free to explore most of it even though your current mission tells you to go elsewhere. There are some closed doors/walled off areas that only open up when going there on an active mission. When entering those mission areas, you enter your own instance (with any group members if you have them). In the open world, you can encounter other players just running around too, they vanish if they enter one of their mission areas.
I think most worlds will probably come with a "explore" mission, like Earth has. This lets you roam most of the map, picking up small missions as you go. Kill x enemies, gather x items (from killing enemies) or scan area/item. Very run of the mill tasks.
PvP areas also share the same maps, the difference between the missions is basically the waypoint will point you to different areas of it.
Now, as to the rewards for going off the beaten path, they are fairly minimal. You can find dead ghosts (the little robots) to revive which gives you some points, or find chests with some money/items. However, as I said, most boss battles and items associated with them (reward as end of mission, not by drop of the mob) are tied to the missions and locked off. Even though you could explore a large area on the moon, you only had one mission. No bosses or drops other than the random loot that drops from any sort of enemy.
My experience seems to suggest very little, if any, incentive in exploring off the beaten path on regular missions in terms of loot. Most areas will be explored by doing the missions in order anyways. Chests/ghosts are best left to be found on the "explore" missions as you then can pick up the tasks along the way for getting extra rewards along the way. The worlds special events can activate on any mission, they all happen on set locations, no random events this far at least.
Note: this of course is from the early level planets. Maybe at higher levels there will be entire planets that are just "explore", where the boss battles are not locked off, and actually drop loot. Those planets might be more enjoyable, but I do share the concerns others have raised here about their very tight lipped approach to how much content there actually is.
