Lost here your saying that the same keyword search is giving different search results depending on the browser
Mind posting a screen shot of this.
How much different are they like a whole different top 10 or just from time to time one site changes order.
It could be that google is doing a test by reading your browser and sending sites that best fit Chrome just to make it appear to be better over FF for speed and stuff.
I have also heard that in different parts of the country you get different result for the same keyword search.