My name is***** and I'm happy to help with your question today. I need to ask you some preliminary questions to be able to help you.
how much is the difference between the quote and estimate?
The additional is for parts - which was not clearly identified as not included in the quote
I see. And presumably you have received those parts?
Not yet - these are for a repair
I was trying to avoid claming on my insurance for an accident I had by paying for the repair
There is no legal definition of a "quote" or "estimate". We have to apply an everyday meaning. A quote is understood by all as a an agreement to provide a service for a fixed price. An estimate is flexible and is to give a general idea. You can hold the seller to the quote as that was the basis of the agreement between you. But what that means is that the seller does not have to supply the addiitonal service or parts which were not mentioned in the quote.
The quote does state Supply and fit of parts,
It could be a mistake. And by the sounds of it they were giving you an estimate not a fixed quote. It would be difficult to enforce an agreement based on a genuine mistake.