To expand a wee bit on Andy's admirably concise answer ...
If you buy from a dealer, you have the protection of consumer law, including (if you pay part or all on credit, the ability to claim from the credit provider in any dispute). Depending on the dealer, they may also have a reputation to protect. Possibly even a manufacturer's back-up.
Buying privately, you have none of that. You have no come-back whatsoever, unless the seller actually misrepresents something. There's plenty of advice available online on what you need to check, but your diligence needs to include as a minimum a credit check (is the car his to sell), MOT and service history, condition (get an expert involved if you aren't one) and a test drive (for which you will probably have to buy insurance). Also, make sure the "private" seller is not in fact a dodgy dealer.
Your choice, but for your sort of money I'd be going to a dealer every time.