I read it as a short story in "Analog" magazine -- back in 1977, if I remember correctly.
The short story is better (imho)—more to the point. The book felt like a short-story that was filled out to make a book.
I believe the author has made the short story available for free on his web site.
The Wiki page on the book backs you up on this - and if I remember correctly, writing the book form in 1985 allowed Card to add in the whole videogame angle. And then he updated it again in 1991, to reflect political facts (according to the Wiki - I'm willing to guess he wanted to erase references to the Soviet Union).
"Ender's Game" isn't the only Sci-Fi novel that grew out of the magazine space Off the top of my head, "A Canticle for Liebowitz" also did that.
Card wrote this story three times
1) He wrote the novelette for Analog. I read it when it came out, and admired it greatly.
2) Some time later, his publisher asked him to write a sequel. He realized that he needed more background, so he wrote the full novel, then wrote “Speaker for the Dead” as a sequel, dealing with the consequences of Ender’s actions.
3) Some time later, he decided to write about what happened to the earth after Ender’s victory. To start that off, he wrote Ender’s story from Bean’s point of view.
I enjoyed each of these individually, and admired the fact that he could tell the same story three times and make it fresh each time.