Years ago I was buying a piece of furniture and was offered a "don't pay for one year" deal. I innocently asked whether I could get a discount if I paid right away. The answer was a firm no and my attempts to continue negotiating failed to lower the price. It wasn't until years later that I learned why.
If you fail to pay the full amount on time, you get hit with high retroactive interest back to your purchase date. If enough people fail to pay on time, the zero interest for a year deal can actually be profitable for the finance company.
To make things a little more concrete, I looked up a major retailer's don't pay for a year deal. At this store, if you pay on time, there is no interest and no extra fees of any kind for a year. If you are short one penny when the year is up, you pay 12 months of 2.4% interest (32.9% for the year after compounding) on the entire purchase.
If the retailer just offered regular financing, let's say that the interest rate would have to be 8% per year to cover the retailer's overhead plus the expected default rate. This is only about one-quarter of the 32.9% rate that you would have to pay if you get hit with retroactive interest in the don't pay for a year deal. So if more than about one-quarter of people are unable to pay after a year, the financing deal is profitable on its own.
Another possible concern for the retailer is that offering a discount in lieu of the zero percent financing deal undermines the advertising claim that the financing is really at zero percent. However, car companies seem able to walk this line without difficulty.