I see a couple of others have wanted to describe the business model you described of paying up front as fast food. However, I am aware of restaurants which are not fast food that also have the model of paying up front. Therefore, I don't think that's an adequate description.
Probably the best description I can think of is prepay business model. There are many variations - some restaurants are experimenting with buying a ticket in advance for the full value of the meal, just like you buy a ticket if you fly on an airplane or go to a concert. Most cafeterias require customers to pay before they eat, as do many delis and coffee houses (and, of course, most fast food restaurants).
Another word that come to mind is takeout business model. Actually almost any restaurant will offer a takeout service even if it is primarily a dine-in/sit down restaurant. Many also have delivery service in at least a limited area. Some of the reasons why are explained here.
In general the advantages of these kinds of approaches are greater certainty that customers will not make reservations and fail to show, improved table turnover (up to 80% improvement when comparing a prepay model with a post-pay model), and less delays for customers (no waiting for the check after the meal, etc.)
On the other hand a traditional, dine-in/sit-down restaurant with a post-pay business model is trying to upsell customers. The longer a customer sits, the more food, desserts and drinks the customer might order. Some restaurants make up to 75% of their income from selling drinks but a more typical number is probably around 20%. A typical restaurant will charge prices such that the actual costs of the restaurant are: 50% of what they charge for meats, 30%-50% of what they charge for wines, 15% of what they charge for pasta and salad, and less than 10% of that they charge for desserts. So if customers will stay longer and order more foods like desserts, that's almost pure profit to the restaurant.