This is something that happens a lot in the law. Two things that look very different on the surface have similar qualities that the law has picked out as being the important qualities. In that sense, the regulation of lotteries and the regulation of pyramid schemes may not have a lot in common, but what they do have in common is the important stuff. We'll see this come up again when we talk about securities regulations.
In 1889, Judge Sherwood of the Michigan Supreme Court put it best:
A lottery is a scheme by which a result is reached by some action, or means taken, in which result man's choice or will has no part. and which human reason, foresign, sagacity or design cannot enable him to know or determine until the same has been accomplished. (People v. Elliot, 74 Mich. 264 Mich. 1889)A pyramid scheme of the MMF variety shares this trait. If I send a MMF chain letter to 1,000 people, whether or not I get any money out of those people does not depend on anything which I do. It depends solely on the whim of the victim.
This is also why many scammers will cry to the heavens that their scheme is legal because it has a product of some kind attached to it. In that case, a sale depends on the quality of the product and my salesmanship, right? Not just on chance?
Wrong: even if that's true, courts have said that if the product is just a way to hide the pyramid nature of the scam, the product doesn't keep you from being liable. (See the USPIS page on chain letters. The stuff about the product is about halfway down the page.) And besides, even if the scam ISN'T a lottery, regulation of lotteries is only one of many avenues that legislatures have taken to restrict pyramid schemes. It's like claiming that your counterfeiting business is legal because it doesn't use child labor in sweatshops. That's nice, and if you did use child labor in sweatshops, you could be busted for that too, but the crime is still the same.
You can read the earliest American case which I found which explicitly deals with chain promotional schemes as a lottery. Note that the scheme in question actually contains a product - in fact, it contains a vast number of products! The case is from 1900.
As it turns out, the lottery regulation laws are just not used very much anymore to combat on- or off-line pyramid schemes. There are much better laws that have been passed since the lottery regulations that do a much better job of smacking around the pyramid schemer, including mail fraud, securities fraud, and consumer protection statutes. These carry heavier consequences and are easier to prove. Scammers beware!