Hi Mehrdad,
Technically they both mean the same thing and the differences are very slight and differ by the level of formality/politeness
I feel like - this is a much milder/polite way of saying you would like something and may mean that you feel like eating an apple but you could also eat an orange if there were not any apples, for example:
Question: I have apples and oranges here, what do you feel like eating?
Answer: I feel like eating an apple.
I want an apple is very direct and expresses a need or desire i.e.you really want to have an apple. I want would be used with people that you are familiar with.