The first question asks what particular car (any make or model) would you like to buy, and the second question asks what kind of car (a specific make or model) would you like to buy.
This is summary of your sentences.
What car would you like to buy? = You would like to buy what car.
You | would like | to buy what car
Subject | verb phrase | noun infinitive phrase and direct object of the verb phrase
what is the direct object in the infinitive phrase, and car is the objective complement referring back to what.
: which thing or things
What kind of car would you like to buy? = You would like to buy what kind of car.
You | would like | to buy what kind of car
Same as the first sentence, except now the direct object in the infinitive is kind, which is modified by the adjective what and prepositional phrase of car.
: a particular type or variety of person or thing