Is the SQL Server service running under a domain user account? If not, then it has no means to authenticate to the network share. To correct, you should change the SQL Server service to run under a domain account that has access to the network share. When you login via Windows Authentication, the stored procedure runs under your credentials which have access to the network share.
ADDITION Changing the SQL Server service account.
- Run the SQL Configuration Manager
- Click on SQL Services
- Right-click on the SQL Server serice and choose "Properties..."
- Change the account name to something like
<Domain>\SQLUser
I would recommend having a dedicated account on the domain for SQL Server with a long password that is kept safe somewhere. The configuration manager should put the user in the appropriate Local User group which will be named something like "SQLServerMSSQLUser$$MSSQLServer" which will give it the necessary rights to run SQL Server as a service and access the files and folders it needs.
For more information:
Setting Up Windows Service Accounts