Judging by this test footage, the answer is "a lot."
A video posted to YouTube by a user named Bennett Sorbo (who appears to be a software engineer at Microsoft) shows side-by-side performance comparisons between an iPhone 6S before and after its battery was replaced. The difference is stark, and most obvious when trying to load apps such as Spotify, as well as web pages, as the iPhone with the new battery is much faster.
By the end of the video, the iPhone 6S with a new battery finished all of the tests in 4 minutes and 33 seconds, 1 minute and 12 seconds less than it took the iPhone with the older battery (5:45). The iPhone with a new battery also scored higher on the Geekbench 4 general performance test, with a score of 4,412 that embarrasses the 2,485 from the iPhone with the battery in need of replacement.
According to 9to5Mac, Sorbo says his device was originally purchased two years ago and was being performance throttled prior to the battery replacement. (If this Bennett Sorbo is the same person that works at Microsoft, his interest in performance is no shock, as that Bennett Sorbo spent time on the Direct3D team working to help developers improve the speed of their games on Windows.)
If your older iPhone has shown signs of aging, watch this video and see for yourself how much $29 (the cost of Apple's own battery replacement program, if you're out of AppleCare) can do for you. The only issue is that you'll be waiting (an average of two weeks) for that appointment, and reports from Barclays analysts show that this wait time has only gotten longer.
This is due to a battery shortage that now requires employees to order batteries when the customer reaches out, and then contact the customer to come in when the batteries arrive weeks later. iPhone 7 batteries are the most likely to be in stock, while iPhone 6 Plus batteries may not arrive until March or April.
Fortunately, there are also third-party battery replacement services for the iPhone, which we've broken down for you in our comparison.