No, they are essentially the same. When I was at Duracell, we private labeled to Ace Hardware, Sears, Costco, Coast to Coast, True Value, Safeway, Albertsons and many other retailers....other than the battery jacket and package, they were identical product quality and performance. We asked "why not make them with less chemical or performance to protect the Duracell brand?" Our R&D team said when you start messing with what is already an optimized design, even lessening the chemistry could induce unintended quality concerns...leaking or some other problem. So they only manufacture to one quality and performance standard. Most private label contracts for retailers were swapped back and forth between Duracell and Eveready who were very cutthroat against each other for those agreements.
We expected the same from our partners that private labled to us. Panasonic (many years ago) made a lot of lithium products for Duracell. That packaging was the same Duracell you would expect to see, as were the part numbers, but instead of "made in USA" they would say "made in Japan". We never knew what product our customers got, we shipped side by side USA and Japanese product. Duracell (at the time) simply did not have the capacity to meet customer volume demands and needed another supply source.
I know in the last decade or so, that all the private label agreements Duracell and Eveready had for retailers appear to have ended with the exception of Costco Kirkland Signature (still Duracell). All the other retailers use off shore battery manufacturers in China, Indonesia and Malaysia. Just read the package "made in"

Varta used to private label for Ikea for decades, but I noticed a few years ago Ikea batteries are made in Asia now too.
You can figure out a lot from looking at what country the battery is from, construction shape and style and packaging to determine who made it.
For low volume selling products like watch or camera batteries, I noticed Duracell is going private label and getting them from other sources rather than make their own. Many of their watch batteries say "made in switzerland" which means probably made by Renata. Which is a very respected European watch cell manufacturer.
It was a long time ago, but when I was there it was rumored we made some camera batteries for Eveready. Companies do that a lot. I aways questioned "why help your competition", but the CEO of a medical company I now work for, said "its just good business sense". "As long as it is profitable, we will sell to our competitors...if we didn't, someone else will. While there are secrecy agreements, we can still glean some competitive information on their sales volumn based on how much product they buy from us."