Please use the 4th parameter of moment.isBetween function (inclusivity). Example:
var startDate = moment("15/02/2013", "DD/MM/YYYY");
var endDate = moment("20/02/2013", "DD/MM/YYYY");
var testDate = moment("15/02/2013", "DD/MM/YYYY");
testDate.isBetween(startDate, endDate, 'days', true); // will return true
testDate.isBetween(startDate, endDate, 'days', false); // will return false