Seeking QGIS equivalent of ArcGIS Erase tool?

Yes there is.

It's called Difference, and you can access it by doing Vector > Geoprocessing Tools > Difference.

In ArcGIS Desktop you can Erase without purchasing the Advanced level license by activating ETGeoWizards in a standard ArcView license.

If you right-click on your toolbar you will get a list of tools, click on "ETGeoWizards" and it will activate that tool.

After opening ETGeoWizards tool, on the left hand side there are a number of categories choose the "Overlay" cat. and "Erase" is 3rd down from the top.