From its very name, we can infer that a business "model" does not refer to the business itself, but to a model that mimics the business. A business model and an actual business are naturally different, just like a miniature plastic model of Gundam is different from a life-sized Gundam.
Why then do we expressly make business models? The reason for this is that trying to understand a business in its full-size is too difficult. If it was Gundam that we were talking about, we could still look at the full picture if we tried, but if we were talking about a huge building, this would be impossible. Even if we looked at it from far away, this would still be a one-sided view which would not account for having gotten a grasp of the whole picture.
It is because of this that, if we really want to try and understand the full picture, we have to look at a scaled down version and not at the actual thing. If we have a model to look at, it does not matter how big the actual building is - we can look at it in its totality.
By looking at the full picture in this way, the flow of the management resources such as people, goods, capital and information, becomes easily visible. If we look only at the sales department we will not know about any problems that may be going on in the production process, in the same way that those in the production site cannot possibly be aware of any dissatisfaction that the customers may be experiencing. Solving management problems is a task that the whole company should be involved in, instead of breaking the problems down into smaller units for each department to deal with. A business model is essential in order to do this.
Understanding a business in its totality is not necessary only when setting up a new business. In the same way in which, once a building has been built, we tend to look only at the most visible parts that we see every day, once a business starts rolling we tend to lose sight of the whole picture. In those instances, we should look at it again as a model. This is why it makes sense to use business models even when dealing with existing businesses.