Just to point out, prior to 6.0, you have to do quite a bit of security configuration to get the file browser working when you use Windows authentication in SQL Backup's UI and are running the UI from a computer other than the SQL Server and the share was on a third computer. This situation causes a "double-hop" authentication issue that you can only work around by ensuring that you authenticate to the SQL Server using Kerberos.
Versions 5.3 and lower simply require you to be a member of the computer's local administrator group, then the file browser would use the SQL Server account credentials. It's a bit messy, I'll admit. That's why the "browsinguserlist" registry key came into being.