I just submitted my own FLAS applications. My FLAS statement and the statement of purpose I submitted with my graduate application covered the same topics, but I tried to write the FLAS statement from a slightly different perspective:
I went into more detail about the personal significance of the target language to me (since it's my heritage language).
I didn't describe my research interests in as much detail as I did in my SOP, because I figure that a FLAS statement will be read by a wider audience than the specialists in the department I applied to. Instead, I concentrated more on summarizing my interests and relating them to a more general academic discourse.
I also articulated several specific ways in which my research will benefit from knowledge of the FLAS language.
I didn't copy-paste anything from my SOP; I wrote a new FLAS statement. I figure that if there is any overlap between the FLAS committee and the admissions committee, those individuals will just get to gain an even better understanding of my background and academic goals.