Both have to do with things that the a speaker wants to do, but they have quite different meanings-- In many cases "I would like" is a more polite way of saying "I want." In restaurants, for example, we often say "I would like the soup of the day...etc." It can also be a way to express plans or desires, like "I would like to travel to Spain one day" or "I would like to be a better dancer." "I would rather" is about preferring one thing over another: "I would rather work ten hour days and have three day weekends than work a regular nine to five" "I would rather eat pizza than pasta." "I would rather travel to Spain than Italy." Often the formula "I would rather [first thing] than [second thing]" is used, but it can still be used when of the two options is implied. A: Do you want to go to the party later? B: I would rather not. A: Would you like to spend the weekend in the forest? B: I would rather rest at home.

Here is an example: "Would you like to buy this computer" a. Yes, I would like to (buy it). b. No, I'd rather buy that one.