Pyramid schemes and Ponzi schemes share many similar characteristics in which unsuspecting individuals are fooled by unscrupulous investors who promise extraordinary returns. However, in contrast to a regular investment, these types of schemes can offer consistent "profits" only as long as the number of investors continues to increase. Ponzi and pyramid schemes are self sustaining as long as cash outflows can be matched by monetary inflows. The basic difference arises in the type of products that schemers offer their clients and the structure of the two ploys.
Ponzi schemes are based on fraudulent investment management services – basically investors contribute money to the "portfolio manager" who promises them a high return, and then when those investors want their money back they are paid out with the incoming funds contributed by later investors. The person organizing this type of fraud is in charge of controlling the entire operation; they merely transfer funds from one client to another and forgo any real investment activities.
On the other hand, a pyramid scheme is structured so that the initial schemer must recruit other investors who will continue to recruit other investors and those investors will then continue to recruit additional investors and so on. Sometimes there will be an incentive that is presented as an investment opportunity, such as the right to sell a particular product. Each investor pays the person who recruited them for the chance to sell this item. The recipient must share the proceeds with those at the higher levels of the pyramid structure.
There are two additional important factors to consider: the only guilty party in the Ponzi and pyramid scheme is the originator of the corrupt business practice, not the participants (as long as they are unaware of the illegal business practices). Secondly, a pyramid scheme differs from a multi-level marketing campaign which offers legitimate products.
For related reading, take a look at What Is A Pyramid Scheme? and The Ghouls And Monsters On Wall Street.
This question was answered by Arthur Pinkasovitch.