επειδή means "because"
Δεν ήρθα επειδή ήμουν άρρωστη. = I did not come because I was sick.
The main meaning of the word "γιατί" is "why".
Γιατί άργησες; = Why you were late?
γιατί also means because when we speak informally.
Δεν ήρθα γιατί ήμουν άρρωστη. = I did not come because I was sick. (γιατί here is more informal than επειδή in the sentence above.)
However, γιατί can not replace επειδή in the beginning of a sentence.
We can say "επειδή ήμουν άρρωστη δεν ήρθα", but we never say "γιατί ήμουν άρρωστη δεν ήρθα".
The only case that γιατί is used in the beginning of a sentence when it means because is only when we give a short answer straight after a question.
- Γιατί δεν ήρθες; = Why you did not come?
- Γιατί ήμουν άρρωστη. = Because I was sick.
But it is wrong to say:
- Γιατί ήμουν άρρωστη δεν ήρθα.