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Obviously the amount of the down payment effects your monthly payment but does it have any bearing on your rate if you have a score of around 685 - 700?
Answer may varies depending on the financial institution, but typically putting down a minimum of 20% will get you the best rate. My credit union (BECU) quoted me about 0.10% higher should I not put down 20%.
For factors, check out this thread: http://ficoforums.myfico.com/t5/Auto-Loans/Factors-Affecting-Auto-Loan-Rates/m-p/1310037#M20635