pretty much a box mod is mod that's shaped similar to or like a box. I know it sounds simple, but the box mod design allows the electronics to be housed beside the battery, and thus reduces the overall height of the mod. for mechanicals, a box mod allows the switch to be on the side instead of below. or above the battery. The tube mods offer a slimmer, albeit taller solution. most people prefer box mods because it's easier to pocket without having half your mod stick out of your pocket. There are other reasons, but those are the main ones.
Now, with the MVP 3 pro, it has the battery built in. That means that when the battery dies, it will have to be plugged in. With the subox design, it has a battery compartment that you can put your own battery in. With the subbox, if your battery is starting to run low, you can swap it out for a freshly charged one. It also gives you the option to charge on the device like the MVP. The downside is, you have to supply your own battery and charger to charge separately. Make sure you get good batteries from a trusted vendor... Sony VTC4's are the safest at the moment.