Mathematica 12.1: Don't use Accumulate or Differences on Around objects!

Okay, wow. This stumped me for a bit.

But I found the answer. It seems to be a bug. An innocent looking one.

The fact you tried Integers instead of Reals triggered it. (I tried to use Reals first and failed to reproduce)

What happens is the following. Whenever Accumulate sees two equal values, it multiplies them by two instead of adding them. Which makes sense for reals

Accumulate[{a, b}]
Accumulate[{a, a}]

{a, a + b}

{a, 2a}

But not for Around's. The multiplication by two yields a different error propagation in the standard linear Gaussian approach

Accumulate[{a,b}]/.{a->Around[v,d],b->Around[v,d]}
Accumulate[{a,a}]/.{a->Around[v,d]}

{v±d,2 v±Sqrt[2] Sqrt[d^2]}

{v±d,2 v±2 d}

Judging from the comments, seems to be a bug. Someone/You should file it with Wolfram.

12.1.0 for Microsoft Windows (64-bit) (March 14, 2020)

Yes, this seems to be a bug. Compare, for example:

list1 = {Around[1., 2.], Around[1., 2.0001]};
Plus @@ list1
Fold[Plus, 0, list1]
Last @ Accumulate[list1]

Around[2., 2.8284978363081703`]

with

list2 = {Around[1., 2.], Around[1., 2.]};
Plus @@ list2
Fold[Plus, 0, list2]
Last @ Accumulate[list2]

Around[2., 2.8284271247461903`]

Around[2., 4.] (* result from Accumulate*)

I reported the issue. In the mean while you can use:

FoldList[Plus, 0, list2]

to get a correctly accumulated list.