Asking questions right at the outset will definitely go a long way towards helping you understand your customers better. It will also help you provide a service they consider to be valuable.
As the article TG relayed suggested, if they're switching, asking them why they're switching will give you a good indication as to the areas you'll need to pay special attention to if you happen to land the account. After they've given you their reasons, tailor your sales message as best you can to highlight your experience, expertise, or service policies related to their concerns. Don't make any empty promises and always makes sure you follow through on any promises you do make.
If they're purchasing services for the first time, ask them what prompted them to buy. Again, their responses will give you good insight into what areas you should pay particular attention to. It will also help you understand potential customers that are very similar. Often times, the reasons they've decided to begin purchasing services will be the same reasons shared by others. Take note of those reasons and use them to your advantage.
Finally, as Joel mentioned, if you can tell someone's 'price shopping', don't waste your time. They'll end up being more trouble than they're worth and your time is much better spent finding and developing relationships with customers that appreciate your work for more than just the amount you charge. If you're competing on price alone, you're asking for more headaches, hassles, and hard-times than you know!