Is there a difference between "would like" and "would want"?
Sort of... it's sort of a connotational difference.
I would like to visit Vienna.
If I were in Vienna I would want to eat chocolate, lots of it.
"would like" is sort of abstract, it's not really anything definite it's just sort of broad and vague in it's meaning. It's something you would like to do or not do, to have or to not have.
"would want" is slightly more situational. Were you in a certain circumstance or situation you would want to do something, but until that circumstance is realized (in this case being in Vienna) it's not really worth mentioning.
I would like to own a Jaguar
If I owned a Jaguar I would want to race all my friends.
So there is sort of a difference between the two, but it's not much of a difference at all. It's just enough of a difference to make the two phrases slightly awkward if used interchangeably.