A significant difference in download speed from one browser to another isn't something I've experienced, but I haven't tested using Safari. From my limited testing, it appears that the upload speeds reported by a given speed test tool will be consistently higher for one browser than the same test with a different browser. It may be the speed test tool you used reports a higher download speed when using Safari than when using Chrome. Which speed test did you use? Were your results from a single test for each browser or a pattern determined from multiple tests for each? Did you select the same city for both browsers? Would you be willing to use a different speed test tool to test your speeds with Safari and Chrome and post your results? If you do, be sure to select the same city for both and perform the tests as close to the same time as possible to eliminate variables.
As for suggestions regarding your wireless, there are others in this forum who can better help you with that. You'll need to provide them some detail information about your setup.