"would like to" is more common and is a standard way of expressing a desire to do something. "would want to" feels more tentative or hypothetical. It might typically be used when describing an imagined situation rather than expressing a direct wish: "If I was boss of this company I'd want to know exactly where the money went"; "I don't think I'd want to spend a whole weekend there".
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.