Perform a Case insensitive Like query in a case sensitive SQL Server database

In addition to using lower(), you need to apply it to the column not the pattern. The pattern is already lower case.

Select  top 300 a.*
from (SELECT userNo, userName, Place, Birthdate
      FROM usertable 
      where personid = 2 and lower(Name) LIKE '%john%'
      UNION 
      SELECT userNo, userName, Place, Birthdate
      FROM usertable2 
      where personid = 2 and
            (lower(Name) like '%john%' or lower(Place) like '%ny%')
     ) a
order by userNo;

Note that UNION ALL is preferable to UNION, unless you intentionally want to incur the overhead of removing duplicates.

You can use UPPER or LOWER functions to convert the values to the same case. For example:

SELECT *
FROM YourTable
WHERE UPPER(YourColumn) = UPPER('VALUE')

Alternatively, you can specify the collation manually when comparing:

SELECT *
FROM YourTable
WHERE YourColumn = 'VALUE' COLLATE SQL_Latin1_General_CP1_CI_AI