The difference is the Microsoft.SharePoint.WebPartPages.WebPart base class is meant for backward compatibility with previous versions of SharePoint. The benefit of using the SharePoint WebPart base class is it supported:
Cross page connections
Connections between Web Parts that are outside of a Web Part zone
Client-side connections (Web Part Page Services Component)
Data caching infrastructure
ASP.NET 2.0 WebParts are generally considered better to use because SharePoint is built upon the ASP.NET 2.0 web architecture. Inheriting from the ASP.NET 2.0 base class offers you features that inherit to ASP.NET 2.0, such as embedding resources as opposed to use ClassResources for deployment of said types.