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.
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.