M/T is Mud Terrain and A/T is All Terrain, standard industry designation for tire types. Firestone may brand their Mud Terrain as 'Max Terrain' but that's just for advertising, or possibly to get across that you can use it for more than mud. Lots of dedicated off-road rides use mud terrains for everything from rock crawling to baja racing.
You should equip your vehicle based on the use it normally sees. There's no point in dumping money into a daily driver to make it better at mud bogging if you will just do that once in a great while, and the rest of the time use it as your daily driver with almost all paved road use. Mud tires will waste gas and are usually pretty noisy, too.
The Explorer is pretty capable off road, especially the short-wheelbase Sport, and with 4 wheel drive. All-terrains will give you the best of both worlds, both the off-road traction when you need it, and the smooth, safe pavement manners on-road.
You can still do mild mudding in a stock Explorer with all-terrains, just don't try to go through thick mud that's higher than the bottom of the bumper, or through water much higher than the tires. Really its best not to go into water deeper than the bottom of the body or doors if you want to stay dry and not have to air out the carpet or put up with a nasty smell.
Whatever tires you use, you should still invest in some recovery gear. It'd suck to get stuck, then have no way to get it out. Even worse, having to go get the gear or a tow...and come back to find your ride stripped, or that someone has pulled it out for you..all the way to wherever they took it. Sucks but it does happen.