It is really not possible to compare this Edison short with a film such as Birth of the Nation, which had in fact come out, not two years before, but earlier the same month.
Nor is this film especially old-fashioned for what it was clearly intended to be - a short educational film, of which Edison made many during this later period. They were often in fact commissioned films and had a small but important market in schools etc. Such films have in practice not changed enormously over the years and most of them tend still today to seem "old-fashioned" in much the same way.
No doubt Edison was encouraged to make (or requested to make) this film to tie in with the success of Griffith's blockbuster (and both film-makers would obviously have been conscious that it was fifty years since Lincoln's assassination). Nor should one forget that, two years before the Griffith film, Thomas Ince had already made a highly successful full-length feature (sadly lost) devoted to the same period - The Battle of Gettysburg. But here Edison was not in any way attempting to rival these major productions by Ince or Griffith nor even to make what would today be called a "biopic".
This sort of educational film remained popular and, indeed, became even more popular during the twenties after Pathé brought out the first home-projector (the Pathé-Kok later the Pathé-Baby known in the US and Britain as Pathéscope). This kind of short film was exactly what sold well in the home-viewing market.
It is an error to see the history of cinema as some kind of straight line where each film is attempting to do the same thing. The repertory of early cinema was even broader than it is today and fulfilled many different functions, many of which (newsreels, situation comedies, magic shows, educational films) are now more often seen as the province of television. To compare this film with The Birth of a Nation is a bit like comparing a television schools broadcast of today with a mainstream feature film. Compare it more realistically with such a "schools" broadcast of out own time and it is perfectly adequate if nothing very special.