Basic difference- Fatigue is a noun and a verb. Tired is an adjective. The equivalent of Fatigue is tiredness not tired. I think you can use "fatigued" as an adjective. Though i am not completely sure of that.
Although they mean the same thing, the way it is used in a sentence is different.
You can say "I am so tired because i worked all day."
You absolutely cannot say "I am so fatigue because i worked all day."
But I've heard people say "I am so fatigued...."
Fatigue is usually used when talking about medical conditions.
You can say," She used to work all day and night. This led her to suffer from fatigue. Poor girl. She must be tired."