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: 
Screen readers assume the content inside a `table` is tabular, and reads it as such. E.g. "row 1, column 1: (contents)". If you use tables to lay out your site, this won't necessarily make any sense. You are telling the end-client you have data with tabular significance, when you actually don't.

By contrast, `div` have no meaning other than "section", so screen readers make no attempt to signify them. You can use divs to make arbitrary visual breaks in your layout without impacting the *meaning* of the markup.

This is what we mean when we say "semantic" markup. Semantic means the markup accurately describes the *meaning* of the content inside of it - tables wrap tabular data, `UL`s wrap unordered lists, etc.
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.