They are very similar in sleep quality depending on your preferences, but that's about as close as you get to similar when it comes to sheets, Egyptian cotton and bamboo.
Bamboo sheets are not legally or scientifically still bamboo. The name of bamboo sheets is not really being honest. Bamboo sheets as we know them today are nothing more than rayon or viscose that has been produced from bamboo and it's the only legal way to advertising them.
If you pay attention to actual product descriptions on sites that sell bamboo sheets, they all either have the word rayon or viscose in their descriptions.
Egyptian cotton, if you get the Giza line, are exceptional quality and you can tell by the difference in prices. A single sheet of true Giza 45 Egyptian cotton could set you back 700, 900 or even 1,300 dollars, and that's just a top sheet. A complete set of Egyptian cotton bedding could drain your account of 4,000 dollar all the way up to 8 ,000 dollars.
Bamboo sheets will last 3 to 7 years on average, a quality set of Egyptian cotton can last decades, and you could pay as little as 250 – 450 dollars for a set from the Giza line, just not Giza 45.
Egyptian cotton, because of the extra long fibers makes stronger thread, making them one of the lest prone sheets to pill over time.
There is a lot more to the topic and you can learn more about bamboo vs. Egyptian cotton on Luxury of the Pharaohs, a site mostly about Egyptian cotton. Just make sure you do your homework before buying because bamboo sheets may fit your budget better and you still get the luxurious feel of Egyptian cotton at a fraction of the cost, but will not last as long.