DATEDIFF() computes the number of boundaries crossed for the given segment. So Sep 2 to Aug 31 is two days (crosses day boundaries twice), but will still always be one full month because it crosses the boundary.

This function returns the count (as a signed integer value) of the specified datepart boundaries crossed between the specified startdate and enddate.

There is no way to get it to show 0 instead of 1 if the values indeed cross the boundary, and using smaller segments like day will trip you up at odd times. The only option is to also check the day value of both inputs to decide if you may want to also subtract 1 from the final result.

This is the kind of thing you need.

SELECT DATEDIFF(month,
        CONVERT(DATE, '05/09/24', 5),
        CONVERT(DATE, '04/10/24', 5)
    ) - IIF(
        DAY(CONVERT(DATE, '05/09/24', 5)) > 
        DAY(CONVERT(DATE, '04/10/24', 5)),
        1,0
    );

This produces 0.

If you only care about dates (and not times) you can use the expression:

DATEDIFF(month, date1, date2)
    - CASE WHEN DAY(date1) > DAY(date2) THEN 1 ELSE 0 END

If you care about time and consider the difference between 2024-07-15 16:44:30 to 2024-08-15 16:44:29 to be less than one month (by one second), you can use the following:

DATEDIFF(month, datetime1, datetime2)
    - CASE WHEN DAY(datetime1) > DAY(datetime2)
                OR (
                    DAY(datetime1) = DAY(datetime2)
                    AND CAST(datetime1 AS TIME) > CAST(datetime2 AS TIME
                )
           THEN 1 ELSE 0 END

or alternately:

DATEDIFF(month, datetime1, datetime2)
    - CASE WHEN DATEADD(month, DATEDIFF(month, datetime1, datetime2), datetime1)
           > datetime2
           THEN 1 ELSE 0 END

Each will adjust the month difference down by 1 if the first date/time is later in the month than the second date/time.

Note that for near-end-of-month start date/times where the ending months are shorter, the month will not be credited until the date rolls over to the 1st of the following month. For example:

• 2024-01-31 16:45 to 2024-02-29 23:59 is considered 0 months.
• 2024-01-31 16:45 to 2024-03-01 00:00 is considered 1 month.

Sample results:

Where:

• DiffRaw is the original DATEDIFF() calculation.
• Diff1 is the day-only adjustment calculation from above.
• Diff2 is the day + time adjustment calculation from above.

See this db<>fiddle for a demo.

I would use a case expression and subtract 1 if the seconds date has a smaller day.

DECLARE @date1 DateTime
DECLARE @date2 DateTime

SET   @date1=convert(datetime,'05-09-24',5);  --Italian     5 = dd-mm-yy
SET   @date2=convert(datetime,'04-10-24',5);

SELECT  CASE WHEN  DatePart(day,@date1) < DatePart(day,@date2) THEN
             THEN  DATEDIFF(month, @date1, @date2) 
             ELSE  DATEDIFF(month, @date1, @date2) -1
        END as AbsoltuteDiff

fiddle