Use a built in function to calculate the time difference between the two dates. As Thom A and Vishal Hadawale already suggested use DATEDIFF.

But I think you can throw away the CASE in your query as it will always go to the else, because '01/06'2020' can never be equal to ''.

As you've mentioned that the expected result is 36 as the difference is 35 Months 30 Days.

If you want every month to count as full: (e.g. 02/05/2023 counts the same as 31/05/2023)

DECLARE @start_date date = CONVERT(DATE, '01/06/2020', 103);
DECLARE @end_date date = CONVERT(DATE, '31/05/2023', 103); 
SET @end_date = DATEADD(month,1,@end_date); 

SELECT DATEDIFF(month,@start_date,@end_date) as "DirectDatediff"; --Result: 36

If only end_dates from the 15th upward should count:

DECLARE @start_date date = CONVERT(DATE, '01/06/2020', 103);
DECLARE @end_date date = CONVERT(DATE, '31/05/2023', 103); 
--SET @end_date = DATEADD(month,1,@end_date); --use if every started month counts fully

SELECT DATEDIFF(day,@start_date,@end_date) as 'NoLeapYear in days'; --Result: 1094
SELECT DATEDIFF(day,@start_date,@end_date)/30 'NoLeapYear in months' ; --Result: 36

SET @start_date = CONVERT(DATE, '01/06/2023', 103);
SET @end_date = CONVERT(DATE, '31/05/2026', 103); 

SELECT DATEDIFF(day,@start_date,@end_date) as 'LeapYear in days' ;--Result: 1095
SELECT DATEDIFF(day,@start_date,@end_date)/30 as 'LeapYear in months'; --Result: 36

SELECT DATEDIFF(month,@start_date,@end_date) as "DirectDatediff"; --Result: 35

SELECT '' + Cast(@start_date as varchar) + ' to ' + CAST(@end_date as varchar) + ' (' + CAST(DATEDIFF(day,@start_date,@end_date)/30 as varchar) + ' Months)' ; 
-- Result: 2020-06-01 to 2023-05-31 (36 Months)

DB<>Fiddle

It seems like the existing SQL code calculates the difference between two dates in days and then converts it to months by dividing by 30.0. However, this approach may not be accurate for periods that include leap years.

To handle leap years more accurately, you can use the DATEDIFF function with the 'month' parameter, which considers the number of months between two dates. Here's an updated version of the SQL code:

SELECT
CASE
    WHEN ('01/06/2020' = '' AND '31/05/2023' = '')
    THEN 'N/A'
    ELSE
        '01/06/2020' + ' to ' + '31/05/2023' + ' ('
        + CAST(DATEDIFF(MONTH, CONVERT(DATETIME, '01/06/2020', 103), CONVERT(DATETIME, '31/05/2023', 103))) AS VARCHAR)
        + ' months )'
END;

This version uses DATEDIFF with the 'month' parameter, which considers leap years and calculates the difference in months accurately.

Make sure to test this code with different date ranges to verify its correctness for various scenarios, including those involving leap years.