I am not sure, but could come up with a few ideas that are probably not too far from the truth in some fashion.
First thing is there is a lot of misinformation online about both fabrics, likely started by marketing departments to launch new brands. In fact, you can build a pretty strong case that marketers, lawyers, and shrewd business people are responsible for many of the woes and misinformation over these two fabrics.
Both have their places, both can place you in the lap of luxury, but there are major differences outside of these simple facts.
Bamboo sheets are not actually bamboo sheets, it's a marketing gimmick based on the theory of global warming. What most call bamboo sheets are actually nothing more than rayon sheets. The FTC fined several companies because the chemical process used changes the chemical makeup of the bamboo cellulose to the point that they can't advertise them as bamboo sheets without calling them rayon or Viscose.
Egyptian cotton is not immune. More playing with the rules that are in place to deceive the public for a dollar. Any cotton, even average grade cotton, like U.S. Upland cottons that are grown in the country of Egypt are considered “Egyptian cotton”.
Now, let's throw some fuel on the fire. Something can be advertised as Egyptian cotton as long as it has 5% cotton made in Egypt.
But wait, there's more!
The only cotton that is of any significant worth out of Egypt is the Giza line of cottons. They are what's known as ELS cottons, which stands for extra long staple. The fibers are 30% longer than your average cotton, making it much stronger when made into threads and then fabrics. This thread makes some of the best sheets in the world, but they are not cheap, depending on which Giza you get. Giza 45 is the top dog, the most rare and highly coveted. You could pay upwards of $1,000.00 just for the flat sheet, and some complete bed-sets cost up to and over $8,000.00.
This is the short story of comparison of Egyptian cotton and bamboo sheets. Luxury of the Pharaohs has a nice comparison article on the topic if you would like to dig deeper into the details of these two incredible fabrics, despite misleading information online concerning marketing.