You don't need to apply to_date() to sysdate. It is already there:
select 1
from dual
WHERE sysdate BETWEEN TO_DATE('28/02/2014', 'DD/MM/YYYY') AND TO_DATE('20/06/2014', 'DD/MM/YYYY');
If you are concerned about the time component on the date, then use trunc():
select 1
from dual
WHERE trunc(sysdate) BETWEEN TO_DATE('28/02/2014', 'DD/MM/YYYY') AND
TO_DATE('20/06/2014', 'DD/MM/YYYY');