SELECT CASE WHEN CAST(date1 AS DATE) <= CAST(date2 AS DATE) ...

Should do what you need.

Test Case

WITH dates(date1, date2, date3, date4)
     AS (SELECT CAST('20101231 15:13:48.593' AS DATETIME),
                CAST('20101231 00:00:00.000' AS DATETIME),
                CAST('20101231 15:13:48.593' AS DATETIME),
                CAST('20101231 00:00:00.000' AS DATETIME))
SELECT CASE
         WHEN CAST(date1 AS DATE) <= CAST(date2 AS DATE) THEN 'Y'
         ELSE 'N'
       END AS COMPARISON_WITH_CAST,
       CASE
         WHEN date3 <= date4 THEN 'Y'
         ELSE 'N'
       END AS COMPARISON_WITHOUT_CAST
FROM   dates

Returns

COMPARISON_WITH_CAST   |  COMPARISON_WITHOUT_CAST
Y                         N

Use the DATEDIFF function with a datepart of day .

SELECT ...
FROM ...
WHERE DATEDIFF(day, date1, date2) >= 0

Note that if you want to test that date1 <= date2 then you need to test that DATEDIFF(day, date1, date2) >= 0 , or alternatively you could test DATEDIFF(day, date2, date1) <= 0 .

The simple one line solution is

datediff(dd,'2010-12-31 15:13:48.593','2010-12-31 00:00:00.000')=0

datediff(dd,'2010-12-31 15:13:48.593','2010-12-31 00:00:00.000')<=1

datediff(dd,'2010-12-31 15:13:48.593','2010-12-31 00:00:00.000')>=1

You can try various option with this other than "dd"

I am always used DateDiff(day,date1,date2) to compare two date.

Checkout following example. Just copy that and run in Ms sql server. Also, try with change date by 31 dec to 30 dec and check result

BEGIN

declare @firstDate datetime
declare @secondDate datetime

declare @chkDay int

set @firstDate ='2010-12-31 15:13:48.593'
set @secondDate ='2010-12-31 00:00:00.000'

set @chkDay=Datediff(day,@firstDate ,@secondDate )

if @chkDay=0
    Begin
        Print 'Date is Same'
    end
else
    Begin
        Print 'Date is not Same'
    end
End

Try This:

BEGIN

declare @Date1 datetime
declare @Date2 datetime

declare @chkYear int
declare @chkMonth int
declare @chkDay int
declare @chkHour int
declare @chkMinute int
declare @chkSecond int
declare @chkMiliSecond int

set @Date1='2010-12-31 15:13:48.593'
set @Date2='2010-12-31 00:00:00.000'

set @chkYear=datediff(yyyy,@Date1,@Date2)
set @chkMonth=datediff(mm,@Date1,@Date2)
set @chkDay=datediff(dd,@Date1,@Date2)
set @chkHour=datediff(hh,@Date1,@Date2)
set @chkMinute=datediff(mi,@Date1,@Date2)
set @chkSecond=datediff(ss,@Date1,@Date2)
set @chkMiliSecond=datediff(ms,@Date1,@Date2)

if @chkYear=0 AND @chkMonth=0 AND @chkDay=0 AND @chkHour=0 AND @chkMinute=0 AND @chkSecond=0 AND @chkMiliSecond=0
    Begin
        Print 'Both Date is Same'
    end
else
    Begin
        Print 'Both Date is not Same'
    end
End

I use the below script in my SQL and it is working fine with me:

SELECT
    ID ,
    CASE
        WHEN DATEDIFF(day, ExpectedDate, ActualDate) < 0
        THEN 'Late'
        ELSE 'OnTime'
    END shipmentStatus
FROM orders
ORDER BY ExpectedDate