Structure training has many facets and side trajectories. The most significant in no particular order are, center-line awareness and control, power investigation and development, and creating potent default stances you can fight your way to when you are loosing in a self-defense situation.
But all that aside, the main purpose of structure training is to learn how to give up control of a fight in exchange for taking a dominant position. This is more or less what I was getting at when I named this blog "weakness with a twist." If you can reposition yourself with a structural advantage, having control over the fight isn't that important. You can effectively let your opponent buck and roll while you tap them on the shoulder from behind. It isn't usually that easy to pull off, but it is that simple.
Structure training isn't the whole fight by a long shot, but it is a very important piece.
Contrast this with momentum training. Learning good structure usually involves a substantial loss of power do to the loss of natural momentum. For instance untrained people often throw their shoulder and head into a punch because they intuitively know that it will increase the momentum of their strike. We martial artists often un-teach this inclination right at the beginning because throwing your shoulder and head into a strike will likely land you in a worse position, especially if the punch misses it's target.
Again, the order in which this unwieldy mass of teachings are learned is up in the air, but there is some logic to teaching Momentum after Structure is established.
If structure training is about giving up control to gain position, then momentum training is about giving up both control and position in exchange for adding chaos. The more mass there is barreling through space along spiral trajectories, the more inherent danger. The less momentum there is in a fight the safer it is. If the person you are fighting is focused on defense, he is less focused on hurting you. If your opponent is trying to control or dominate you, adding momentum will likely shift him into a defensive mode. The more defensive he is, the more rigid and predictable he will become. The more experienced you are with the chaos of added momentum, the more likely you are to prevail.
Momentum training increases the power of strikes dramatically, but that's a side benefit. The main purpose of momentum training is to get you to drop the wasted effort of trying to dominate and control. Tigers fighting other animals don't waste effort trying to dominate and control. Those are social concerns. Drop them and you will experience greater freedom of action.
The will to dominate and control arises from the fear of chaos (huntun). That doesn't necessarily make it good or bad, it just limits our ability to see things as they actually are.