I am inclined to say that they are the same book, just produced under different publishers.
When searching on Amazon (in the United States) for "A Game of Thrones: 20th Anniversary Illustrated Edition". I am presented with the two different covers that your searches also provided.
You will notice that the Harper cover is shown second even though it is an exact title match while the Bantam "original" edition is listed shown first.
Digging a bit deeper the Harper edition is listed as one of seventy-one other formats for A Game of Thrones, while the Bantam is listed as its own work.
Beyond that the Bantam edition shows a preview with some of the illustrations, while the Harper edition preview reverts back to a standard preview.
As mentioned in the comments, the blurb on the German Amazon site for the Harper edition and on American Amazon for the Bantam are almost an exact match
Published in celebration of the twentieth anniversary of George R. R. Martin's landmark series, this lavishly illustrated special edition of A Game of Thrones-with gorgeous full-page illustrations in every chapter-revitalizes the fantasy masterpiece that became a cultural phenomenon. [German Amazon]
Published in celebration of the twentieth anniversary of George R. R. Martin’s landmark series, this lavishly illustrated special edition of A Game of Thrones—featuring gorgeous full-page artwork as well as black-and-white illustrations in every chapter—revitalizes the fantasy masterpiece that became a cultural phenomenon. [American Amazon]
Beyond that I see no reason why a book by the same author, same foreword, same maps, same illustrator, and same number of pages would differ between publishers beyond the cover.