As said before, when it comes to prestige the publisher does not have much weight, the journal does.
But I have to disagree slightly with other answers when it comes to exposure: some publishers have their journals available by many more universities in the world than others, thus bringing some more exposure to the work they publish (exposure is not only availability, but availability is the first logical component of exposure). Now that large commercial publishers bundle a lot, sometimes selling all their journals to a whole country, all their journals (however obscure they sometimes are) are widely accessible. However, this does not make a big difference between Springer and Elsevier, who are both big commercial publisher with mostly convergent commercial strategies. Note that open-access journals are of course even better for availability; but the fact that SpringerLink and ScienceDirect (the platforms used to access Springer and Elsevier journals) gather so many titles under one roof makes it more likely that someone will easily find a given article (an open-access buried into the depth of the internet might be available but difficult to find).
Another difference between publishers you didn't asked for but that matters is the quality of publishing work (copy-editing, assistance to editorial boards, etc.) For an author, the work done on the articles themselves is important. If you think it doesn't matter much, then think about the difference between
A. a publisher that sends you galley proofs to be checked in one week, without any indication of what has been changed from your own manuscript (but many tiny and not-so-tiny changes having been made), and
B. a publisher that sends you an annotated version of your manuscript, showing all modifications made to it, together with the galley proof that resulted from this process, so that you can check each modification easily.
The difference is in hours of work, unless you blindly trust copy-editors. And this would not be wise. Unfortunately for you, from my experience and all the testimony I gathered about various publishers, Elsevier and Springer are pretty similar when it comes to service to the authors: they do a pretty bad job.
So in conclusion, I would say that publishers do matter to authors even when they do not want to take any political stance like the Cost of knowledge pledge. But it is hard to find a strong difference between Springer and Elsevier (if you want names of publishers that do a great job, I know a few but for mathematics).