An estimate answers "what makes up the price". A quotation answers "what exactly, for how much, and on what terms". In practice you send both: the quotation with the terms, the estimate as its annex.
What a quotation needs
A plain-language scope; the price with its VAT status; a validity period, typically 14–30 days; a delivery time counted from the start of work rather than a calendar date; payment terms; a warranty; and — most important and most often missed — an exclusions section.
Waste removal, carrying materials upstairs, scaffolding hire beyond plan, site power and water, moving furniture, weekend work: if it is not written down, the client fairly assumes it is included.
Getting past a straight price comparison
A client with three quotations compares one number. The way out is to show what is inside: material grade and thickness, number of coats, preparation, warranty. Offering two of your own options — base and upgraded — also beats being compared against a competitor.
The usual chain is quotation → estimate → contract → handover act → invoice, each referring to the one before.