When using Scan Name as a filter in your asset report you are limiting the report to display the assets that were found in the scan. The asset report is intended to report all information known for the assets...which is why you see all vulnerabilities found even from other scans. The scan name is only providing the asset report with a filtered list of assets. To get closer to what you are seeing in a scan report you could limit the vulnerabilities displayed in the report to only display the vulnerabilities selected in the scan of concern. In the Sections tab of the asset report template click 'Select Vulnerabilities'. If you are using a Vuln Set this gets even easier. You can chose the vuln set in the drop down...otherwise you can manually check the vulnerabilities you would like to see in the report.
Hope this was helpful,