Application not connecting to SQL database after disaster restore.

This is probably something simple except my complete lack of knowledge of SQL etc prohibits me to solve it.

I have a client with "custom application" that was written for them over a decade ago. This application was connecting to SQL database that was hosted on it's own server. This server recently stopped to exist. Server is dead, drives are dead - the only thing that I have is the good backup of database.

I have build (as virtual machine) a Server 2008 R2 with SQL 2008 R2 installed and I managed to restore database from backup.
I gave the server the same name (and same IP) as the machine that died. This new server is build with windows authentication (and I was told that this is how it was previously)
I have setup SQL port exceptions on firewall and tested the connectivity to database on the SQL server itself and from the workstation successfully (I created text file and renamed it .udl and this gave me Database connection tool)

Regardless all this the "Custom App" is not connecting to database - it gives error: "The application is unable to connect to the database server."

I have found that this "Custom app" has a config file that mentions some kind of account and password.

Do I have to do something else to the restored database - recreate/relink accounts?