There is an emphasis in “tout simplement”. For negative meanings, it usually implies a strong disagreement by the locutor/writer.

It’s only a difference of emphasis and/or style. Both are completely correct, in writing as well as in spoken language. I could come up with another expression that uses "tout" in a similar fashion: "tout betement".

Il s’est bêtement trompé de rue.

Il s’est tout bêtement trompé de rue.

and actually, the second one would be quite similar to:

Il s’est tout simplement trompé de rue.

The first one… might put a bit more emphasis, through "bêtement", on the fact that it was a silly mistake, which IMHO is less the case in "tout bêtement".

Note also, that this construction does not necessarily work with other adverbs.

Quite frankly → tout franchement franchement

Quite clearly → tout clairement très clairement