Google won't care what browser you are using, everything else being equal it will give you the same result regardless (with the possible exception of mobile browsers).
Of course, there's a big caveat there - "everything else being equal". Remember that Google knows a lot about you and your browsing habits, especially if you browse and search while logged into a Google account. And it uses that knowledge to tailor its search results to what it thinks will be most relevant to you. So if it knows that you often visit a particular site, it is likely to push that further up the page than it might otherwise have done. Likewise if it thinks that you often refer to a particular geographical area, or your IP is registered to a particular area, it may push pages relevant to that area higher up. So then, when you search on someone else's computer, all those titbits that Google knows about you have suddenly gone out of the window, and Google doesn't know that your client likes to visit that site so it doesn't prioritise it.
What you're actually seeing there is more likely to be the "true" and unbiased result that most people would see. It's good practice for webbies to keep one "clean" browser that you never use to log in to your Google account, where you can search for words or phrases and not click on the links – you just use this to see the natural ranking position of sites untainted by Google's knowledge of your browsing habits.