Question: How much easier is it for screen readers to handle `<div>` based websites as opposed to older `<table>` based websites, and why are they worse?

Answer: 
Table based sites may confuse the screen reader for the type of data that is being presented. Originally tables were used to show a 2D list of data. But with tables being used to fit designs, the readers would have to distinguish the intention of the table.
The viewpoint that DIVs work better with screen readers than tables is missing the real issue. 

Current CSS and HTML standards simply don't provide any explicit tags/indicators to aid screen-readers. Until they do, regardless of whether you use DIV or tables, you have to just hope that your "semantically correct CSS page", or "get the job done tables" work with most screen-readers.

Don't forget, it is equally easy to create an incomprehensible page using DIVs only, as it is using tables.