In boxing, the main goal is to score as many punches to the head or body and evade as many punches as possible. The concentration and commitment are mainly to the upper body without the use of elbows or attack to the limbs. In MMA, by contrast, the main purpose is to overtake your opponent by punching, kicking, grabbing or choking.
Because of the diversity of the two sports, boxing and MMA differ in the methodology of training. In boxing, the training bag simulates a body, and the concentration of punches will be delivered to the head or body; therefore, the bag must be full of cushion to simulate and absorb the punches. Place the boxing bag in an open area so you can move all around the bag to achieve your training goals.
In the world of MMA, the fighter has many more options than a boxer. The MMA fighter can box, but he can also kick, grab and throw down his opponent; thus the strategy in training is to achieve a more diversified style of fighting. Place the MMA bag higher than the boxing bag, allowing you to kick, punch, grab -- and even elbow it.
In MMA, if you are a good boxer or what is called "stand up fighter" by trainers and competitors, your chances of conquering your opponent are far better than if you are just a grapple fighter or floor fighter. Being able to stand up and move by shadowing your opponent gives you the leverage you need at times to find the opening and score punches, versus just launching and wanting to hold your opponent -- leaving yourself open for counterblows.