is a quote binding or can they charge more later

The distinction is legal, not stylistic. A quote is an offer of a fixed price for defined work. Accepting it forms a contract at that price. An estimate is an informed guess and creates no such obligation, which means the final invoice can differ.

When a quoted price can still change

• The scope changes, which is a variation and should be agreed in writing before the extra work is done.
• The quote itself stated conditions or exclusions, such as excluding rock, asbestos or repairs to hidden damage.
• You asked for something different once work was underway.
• The quote had an expiry date that had passed before you accepted it.

What to have in writing

• The word quote, not estimate, if that is what it is.
• A description of the work specific enough that a third party could tell whether it had been done.
• What is excluded, since the exclusions are where the extras come from.
• Whether the price includes GST, materials, waste removal and making good.
• The payment terms, and any deposit, which is capped by law in several contexts.