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: 
It depends on what a table is used for. If a table is used for layout the screen reader has no way of knowing this and reads cell by cell, starting in row one, column one, row one column two, row two column one row two column two, etc which often isn't the way the content is meant to be viewed. If data needs to be displayed in a structured form tables are the best way to go with screen readers since keystrokes can be used to read row by row and column by column. As a screen reader user I find it anoying when data doesn't appear to make any sence because it is either layed out poorly with a table, or a table isn't used at all and some other visual indication is used.
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.