What's the difference between product sponsorship and paid sponsorship, and how should I approach each?

Product sponsorship is the entry level - you get product and put a decal on your car. This is relatively easy to obtain if you're in a professional arena and fill out website forms. Paid sponsorships require much more. You need to become known personally to the company's current partners and representatives. The best advice is to meet sponsors at events (Optima alone attends 365+ events per year), become friends with their current racers, and get your name known through personal one-on-one interactions. A piece of paper (proposal) is not a great introduction - personal relationships are essential for paid partnerships.