I was able to resolve the differences!
The underlying problem is with the Charitable Donations Optimizer. Online it split the donations by person automatically, and didn't carry any amounts forward to next year.
The download version, the optimizer makes me choose one person or the other to claim all the donations. And DOESN'T advise me of a value that I should perhaps carry forward.
So then I used the EasyStep option in the download version, where it lets me pick an amount to claim on the second person (with the balance going to the first person). So I used the amount that the Online version came up with.
Once I did that, the Pension Optimizer on the download version then came up with the same split amount as the online version. Then causing the refund amounts to be identical in both versions.
It looks like the charitable donation optimizer leaves a lot to be desired in the download version.
However, the online version has problems with charitable donations as well. I tried claiming all the donations on one person, and if physically moved all those donations to the first person. But when editing the donation, it doesn't give me an option to move it to the second person. So to change the split, I had to physically reenter half of the donations for the second person, and delete them from the first person. What a pain!
