They are identical: BETWEEN is a shorthand for the longer syntax in the question that includes both values (EventDate >= '10/15/2009' and EventDate <= '10/19/2009').
Use an alternative longer syntax where BETWEEN doesn’t work because one or both of the values should not be included e.g.
Select EventId,EventName from EventMaster
where EventDate >= '10/15/2009' and EventDate < '10/19/2009'
(Note < rather than <= in second condition.)