An estimate is your best judgement of what a job is likely to cost, given what you know before you start. The final bill can come in above or below it. A quote is an offer to do described work for a stated price, and once the customer accepts it, that price is what they expect to pay unless the work changes. Fair trading bodies generally describe the difference the same way: a quote is a fixed price, an estimate is approximate. Calling a document an estimate does not let you charge whatever you like, though. If the final bill lands well above it without warning, expect a dispute, and expect the estimate to be read as a promise of roughly that figure.
In practice
Most trades send both, for different kinds of work. Fault finding is the classic estimate. An electrician chasing an intermittent tripping circuit cannot know in advance whether it is a faulty appliance or a damaged cable in the wall. The estimate says two to three hours at $110 an hour, so $220 to $330 plus parts, and that the electrician will ring before the bill passes $400. A quote suits the job that follows: replacing six downlights at $95 each is $570, a figure that will not move.
The useful rule is to quote whatever you can see and estimate only what you cannot. Many jobs end up as both on one page: fixed lines for the known work, plus an estimated allowance, often called a provisional sum, for the unknown part, with the rate it will be charged at.
Whichever you send, label it. A customer who reads "estimate" knows to expect movement. A customer who reads nothing assumes a quote. For an estimate, put three things in writing: the range, what the final figure will be based on, and the point at which you will stop and ask before spending more.
In Quote Desk
Every email Quote Desk sends carries the subject "Your quote from [your business]". When what you are sending is really an estimate, open the note with that word and the basis of the final bill, so the customer reads it before the total. See what the customer sees.