Why doesn't this work?

This query changes the database:
PRINT DB_ID()
USE DATABASE2
PRINT DB_ID()

The output is:
7
13

This query does not however:

PRINT DB_ID()
DECLARE @DBName NVARCHAR(10)
SET @DBName = 'DATABASE2'
DECLARE @SQL NVARCHAR(20)
SET @SQL = 'USE ' + @DBName  
PRINT @SQL
EXEC(@SQL)
PRINT DB_ID()

The output is
7
USE DATABASE2
7


The question is why? I would like a decent explanation please or solution to the problem :)

Thanks!