Give both. "My rate is $X and I estimate it will take at least Y hours." That way they get an idea of what it will cost in total and also how much it will cost them to make changes.

What I meant exactly is, give an estimate and an hour quote. Let the client know that if you complete the project in less time, you will charge less and vice versa. I've found that in all projects specs change and things take less or more time than estimated. Nothing sucks more than working free because you underestimated a task. Also if you have a fixed price, you will invariantly be motivated to take the quick and dirty solutions everytime. This will not lead to success for the project.

I think knowing your own hourly rate is necessary to benchmark your own tasks and priorities. But yeah, in my experience clients generally want an estimate up front so my rate is more of an internal tool. The art of it all comes from being able to accurately predict the amount of time for the project that will used for the estimate based on my rate, which comes from experience.

Sure, give an estimate, but then stay in close contact - if you're slipping, alert the client as soon as possible.

The keys to billing per project are clearly defined specifications, signed contracts, and down payments. Give your hourly rate in addition for work that goes beyond the spec.

Both you and your client benefit from knowing exactly what they're getting and for exactly how much. Plus, a concrete spec and price will help them get the expenditure approved, when applicable.