Will I have better SEO results if I have Google Analytics on my website, or does Google not care if I'm using their analytics software?
No, you won't get better results simply from using Google Analytics and Google doesn't favour sites with Google Analytics over sites without it installed.
Google Analytics is of course a great analytic tool and when the data is studied correctly, it can influence on your SEO efforts and thus can indirectly affect your SEO. (As could any other types of traffic analytics software)
But simply installing Google Analytics wont give you any kind of SEO 'boost'.
Here is Matt Cutts, head of Web spam at Google, confiming this: