I have added the authorship to a website of mine however it looks like this when I enter my URL into the tool, however when I search for the site on Google, it shows an out of date cached image and doesn't appear like the rich snippet tool. Any insight as to why this is happening?
Rich Snippets only show if you're logged into Google+ or at least Authorship Rich Snippets only show if you're logged into Google+. It also depends what keywords were typed into the search.
Rich Snippets aren't guaranteed, sometimes Google for a keyword phrase might not show them when for a different keyword phrase they will show a Rich Snippet. There is a difference between "Rich Snippets" and Authorship. Authorship is tied to Google+, Rich Snippets are tied to schema.org and a few other standards.
Supposedly Amazon.com doesn't follow any of the recommended standard but Googlebot knows how their website works so they get "Rich Snippets" anyway. It is good to be the king of e-commerce.