US speaker here. They seem like they would mean the same thing and they are sometimes interchangeagle. But there are different shades of meaning and I hardly ever hear 'would want to'. When I have heard 'would want to' it seems like it's either giving advice, basically meaning YOU SHOULD: "You would want to take the A train to get here." or it's about more of a hypothetical activity in the future: "I'm not sure I would want to travel to Mars if I had the chance." For most more standard uses involving concrete plans, etc. you would use 'would like to'. I would like to eat out tonight. Would you like to come visit? I would like to take this class, etc. etc. I'm not sure how official this is - just my observations.

Here are some examples I can think of. A: I would like to win, but he paid me to throw the match. B: If I were you, (then )I would want to win, and I wouldn't do that for any amount of money. "Would want to" is "want to" in the conditional mood. One common use of the conditional mood is in the clause following an "if" clause, like the example above. It was an if-then sentence. I would like to is the conditional mood of "like to". However, "like" is a special case here. People conjugate "like" in the conditional mood as a form of politeness instead of just saying "I want". A common example is ordering food in a restaurant or drinks at a bar. We would say, "I would like...". Of course, you can also use "would like" in the then clause of an if-then sentence. "I thought that if you saw the play, you would like it." "Would like *to*" is just followed by a verb instead of a noun. In a restaurant you might say, "I would like to try the steak." That would be a polite way of ordering a steak. I think perhaps mainly what's confusing you is that we use "would like to" in the context of adding politeness instead of just saying "I want...".

There are many things I would like that I wouldn't actually want. I would like to eat chocolate cake but I wouldn't want to because it would make me feel bad afterwards. When someone offers me a piece, I say "that looks delicious, but no thanks, I don't want any".

‘I would like . .’ is the usual form, and is a polite way of saying ‘I want . .’ It can also express a wish for the future : ‘I would like to go to Taiwan next year’. ‘I would want to . .’ is only used in rather specific contexts, as suggested by Dan, and indicates that either you or someone else should do something.

I would like to is used when you yourself want to do something. I would want to is used when you want someone else to do something so you in turn can do something. E.g i would like to go to the park; I would want him to go to the park before I went to his house