Two ways a thing can stay the same
Mathematical counterpart: Invariance of the action: strict, and up to a divergence.
There are two conditions in circulation. Noether used one of them. The other is usually presented as hers and is not.
The strict condition
Transform everything. Add up the score again. Get the same total.
That is Noether's condition, and it is the only one in her 1918 paper. Nothing weaker appears anywhere in it.
The weaker condition
Now consider a ledger.
You keep a running account through the day. At close of business you compare today's total with yesterday's, and they differ — but the difference turns out to be entirely a transfer that came in at opening and went out at closing. Nothing that happened during the day contributed anything.
That is the weaker condition. The score is allowed to change, provided the change depends only on the two endpoints and not on anything in between.
Why should that be good enough? Because the whole test in this subject is whether a small nudge in the middle changes the score. A term fixed by the endpoints cannot respond to a nudge in the middle. It is there, it is not zero, and it is invisible to the only question being asked.
Who added it
Erich Bessel-Hagen, in 1921, three years after Noether's paper.
This volume states the attribution on every page where the weaker condition is used, and it is not a point about credit. It is a point about what a reader should expect to find if they open the 1918 paper.
Bessel-Hagen also noticed something that gets dropped when his condition is folded into hers. Under the strict condition, if the smallest step works then building it up to a large change also works, automatically. Under the weaker condition that guarantee is gone. So the weaker condition has to be stated for the smallest step and cannot be stated for the finished transformation.
That is a real structural difference and almost every modern statement omits it.
What it buys
One case above all: sliding your whole laboratory at a steady speed instead of leaving it parked.
Under the strict condition, this gives nothing. Under the weaker condition it gives a genuine and useful result — that the centre of mass of an isolated system drifts at a constant rate. See Drifting centres and electric charge.
So the weakening is not a technicality. It rescues a real conservation law that strict invariance loses.
What Noether actually says about divergences
She does make a remark in this neighbourhood, and it is worth separating from the above so the two are not confused.
Take a scoring rule. Add to it a term that only ever contributes at the endpoints. You now have a second rule. Her observation is that the second rule has the same symmetries as the first — though the way those symmetries are described may look more complicated for one rule than the other.
That is a statement about two rules sharing one symmetry. Bessel-Hagen's is a statement about one rule having more symmetries than strict invariance admits. They are neighbours. They are not the same, and only his gives the drifting centre of mass.
Why an endpoint term is invisible to the physics
It is worth seeing why the weaker condition is allowed at all, rather than taking it on trust.
Everything in this subject is settled by one test: nudge the system somewhere in the middle, and see whether the score responds.
Now add to the score a term that is fixed entirely by the two endpoints. Nudge the middle. The endpoints did not move, so the added term did not change. It contributed exactly nothing to the test.
Which means the equations of motion are the same for both scoring rules. Not similar — identical. You cannot tell the two rules apart by any experiment, because they predict the same behaviour in every situation.
So a symmetry that preserves the score exactly and a symmetry that preserves it up to an endpoint term are, as far as the physics goes, equally good symmetries. Only one of them is covered by the 1918 paper.
How to tell which you are dealing with
Transform your scoring rule and look at the difference.
If it is zero, you have a strict symmetry and Noether's own condition applies.
If it is not zero, ask whether the difference is the rate of change of something. If it is, the difference only ever contributes at the ends, and you have the weaker case.
If it is neither, you do not have a symmetry at all and neither theorem applies.
No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.