To get that information the web browser must ask the file system when the file was last modified and compare it. It is recommended to do a "submitted time is less than time now" on the file (which means the date must be parsed), by the RFC (2616 14.25); however, some people do use inequality operators as you suggest.
In this case, it would be the web server that is not following the RFC (sending arbitrary Last-Modified header), not the browser.