First off, let's give props to any prospect who takes the time to tell you that they have chosen someone else. (And double-props to anyone who gives you feedback on why they didn't choose you.)
Flyte's not the least expensive game in town (nor the most expensive), and often it just comes down to upfront dollars for a client. You may be in the same boat.
Many times a prospect has told us that they'd really like to work with us but it's just not in the budget so they're either going to stick with what they have or go with someone less expensive. Sometimes that message is delivered over the phone, sometimes in an email.
If this happens to you, respond immediately. If they left a voice message, call them back. If they sent you an email, hit reply. If you're on the phone with them, don't hang up.
Thank them for their time. Complement them on their choice. (No way you're ever going to recover this prospect if you insult their decision-making capabilities, no matter how boneheaded you think they're being.) If possible, offer them something free (advice, handouts, etc.) that might make their project turnout better.
Most important, leave the door open to future collaboration/work.
I often ask them to please keep us in mind if things don't work out with their first choice, or if they want some help with search engine optimization or Web marketing once the site has launched. I can't tell you the number of times we've ended up doing something with the prospect down the road with this approach.
Other times the prospect leaves with such a positive perception, that they recommend us to associates who end up doing business with us. More than once, a client tells me that so-and-so recommended them, but the name doesn't ring a bell. When I check my database I see that so-and-so didn't do business with us three years ago.
Wow. I guess we made quite an impression.