Are you referring to the missionary free copies of the bom that they pass out? They used to have a hard cover and now have a soft cover. I am sure cost is the reason for this change.
Or, are you really referring to changes that have occured since the original printing of the book and not just talking about the cover of a book distributed for free from missionaries. My personal BOM has a maroon colored leather cover with my name embossed in gold. The inside is the same regardless of what is on the cover.
There have been many covers for the BOM the first one was a hard cover that was a tannish color. then there was the one where Another Testament of Jesus Christ was added. Then there was the one were they had a form of reformed egyptian on it then the sky blue one with the Angel Moroni, then plain blue with an Angel Moroni in gold and finally the ones we have.