My Solution : Failure to connect to SQL Server with provided connection information. SQL Server4 does not exist, ...

Error: Failure to connect to SQL Server with provided connection information. SQL Server4 does not exist, access is denied because the IIS user is not a valid user on the SQL server, or the password is incorrect

Solution:
1. After creating your publication on the management studio, add the user IUSR_ in the security->login folder
Important: you don't have to create that user, this user already exist, so look it up under advanced search!!
2. Rightclick on that user and choose properties.
3. Under usermappings check the box of the database you would like to access, then click ok.
4. Rightclick on your publication and choose properties.
5. Select publication access list an add the IUSR_ user and click ok.
6. Rightclick on your publication, select view snapshot status and generate the snapshot.
7. Execute the web synchronisation wizard the subscription wizard.
this wizard creates the virtual directory for the webaccess, choose annonymous access.
8. Execute the subscription wizard and use windowsauthentication.

Reference:

• http://forums.microsoft.com/MSDN/ShowPost.aspx?PostID=335654&SiteID=1
• MSDN