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.
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.