When talking about a lack of sleep, we also say sleepy. To me, being fatigued is closer to being exhausted, requiring a long period of rest and being tired is easily remedied with a little rest. Fatigued and tired can be synonymous when speaking of a physical state of being weary and/or sleepy. However, we can be tired of our job or tired of being nagged by our parents or in my case, tired of students using translating dictionaries instead of English dictionaries. It means we are almost fed up. :-) I recommend the Cambridge Advanced Learner's Dictionary or similar dictionaries by Longman and Oxford. Using an English language dictionary will not only give you the correct pronunciation and definition, but sample sentences which are essential if you really want to learn how the use the word yourself - integrate the word into your own speech.

They are close, but fatigue means when the body gets tired after running long distances or doing a lot of labor. Being tired can just be not getting enough sleep, but can also be used when talking about the state of the body after doing too much physical labor.