Open-world (level-design) means you can go wherever you want. Sandbox (gameplay) means you can do whatever you want.
This definition probably isn’t 100% accurate, but it should be good enough.
It’s a sandbox, if you are given a quest that tells you to go in a certain direction, and you can say “FUCK THAT!”, turn around and go 4 kilometers in the other.
It’s a good sandbox, if doing the above is fun.
Have a look at Fallout. You could follow the main-story line. Alternatively you could get yourself some supplies, climb on the next hill, find an interesting landmark, travel to it, meet new characters and do other quests for hours.
A bad sandbox would be Mafia 2. Sure, you have pretty much absolute freedom as to where you go next. However, outside the strictly linear chain of missions, there really isn’t anything else to do.
As I already said, open-world is a type of level design. It’s the opposite of linear. CoD has linear level design, while Fallout’s level design is open-world.
CoD works like this:
You go through this hallway till you reach the only door you can interact with. There you will proceed to wait till the super-manly dialogue between your team and HQ has finished. Then you will look at your captain kicking the door open in a very dramatic manner. You will clear that room, jump out of the window and proceed to do a vehicle section that will take you to a specific destination, by a predefined route. And so on…
Fallout works like this:
The destination of your currently active quest is a cave about 2 kilometers north of here. You can stop by, whenever you feel like it. Have fun.
Fallout is also a sandbox, because you can do whatever you want and don’t have to do the current quest right away.