I would suggest that "like" is an ongoing thing. You liked it yesterday, last month, last year, and you will like it tomorrow, next month, and next year.
Well, likes do change - I used to like roller coasters, but now they make me sick.
"Want" is immediate. Although I like the snow, I don't want any to fall right now. I like eggs Benedict, but I don't want some after a big dinner.
Now comes the tricky part:
"I would like" is simply a polite way to say "I want."
"I would like" is an immediate, current desire, phrased more nicely.