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These two words are synonyms. The only difference is that you can not start a sentence with “ama” in written language. You probably hear the word ama more often than fakat in daily speech, especially young people tend to say ama instead of fakat . Also, there is a third word which is "lakin", but it is not that common.
(http://www.learnturkisheasy.com/grammar/conjunctions)
