Nope.You're wrong here.When you close the browser, you cannot get that session again.The idea of storing session on Sql server is the scenario where you have a web farm for your app.At that time even though your one of the IIS server down, still you can access the session which you stored on db by using downed (failed) IIS server.But you cannot do this by using default in process session.
Please read below mentioned article for more info:
How to use ASP.NET session state SQL Server Mode in a failover cluster