MySql - 在varchar的两个日期之间搜索

I have my query;

SELECT ID, 
REFERENCE, 
SHORT_NAME, 
TITLE, 
DATE_FORMAT( STR_TO_DATE( NEXT_EVENT_DATE,  '%Y-%m-%d' ) ,  '%d/%m/%Y' ) AS NEXT_EVENT_DATE_FORMATTED 
  from cases WHERE `CLIENT_ID` = '54' 
  AND NEXT_EVENT_DATE_FORMATTED > '01/10/2014' 
  AND NEXT_EVENT_DATE_FORMATTED < '31/10/2016'

I am pushing data up through heterogeneous services which is why my date column is using a varchar and not an actual date field. What is the correct way to search between the first and last day of this month? (I will generate the where part using PHP most likely, unless mysql knows when the last day of the month is).

Thanks, Harry.