Take out the MONTHS from your case, and remove the brackets... like this:
CASE
WHEN RATE_DATE BETWEEN '2010-01-01' AND '2010-01-31' THEN 'JANUARY'
ELSE 'NOTHING'
END AS 'MONTHS'
You can think of this as being equivalent to:
CASE TRUE
WHEN RATE_DATE BETWEEN '2010-01-01' AND '2010-01-31' THEN 'JANUARY'
ELSE 'NOTHING'
END AS 'MONTHS'