If you have no concern with what tiem the entered_datetime field is just want to see for day then try this.

DATEADD(d,DATEDIFF(d,0,appt.entered_datetime),0) between DATEADD(d,DATEDIFF(d,0,@start_date),0)

                               and DATEADD(d,DATEDIFF(d,0,@end_date),0)