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.
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.