In practice people use these terms interchangeably. Yet it matters for the customer's expectations.
Quote
An offer at a fixed price that the customer can say yes to. After approval there is an agreement.
Estimate
Usually an indication of what something costs. Be clear whether the price is fixed or an assessment.
Budget
A projection of total costs, often for larger work, with a range. Differences with the final costs can arise.
Why it matters
- The customer reads a quote as a commitment.
- An estimate needs explaining what could change the price.
- An unclear term leads to discussion later.
How to avoid misunderstandings
- State at the top what the document is.
- Say whether the price is fixed or an estimate.
- Give the validity period.
- Describe what could cause extra costs.
Legal
The exact consequences differ per situation. If in doubt, ask a lawyer.
Also read the seven things in a good quote.