Google will customize the results it shows you based on the data it has for you. If you are signed into a google account, or have a google cookie, it will give you different results than someone not signed in, or with a different cookie. Internet Explorer and Firefox use different cookies, and you are possibly signed into different accounts, thus explaining the differences.

Also, make sure you are actually using Google in both browsers.

When I test, the person who comes up in Google on Firefox isn't you! Here's a comparison of Google, DuckDuckGo and Yahoo for me. I wonder why?? One of the many mysteries of Google's relevance algorithms.