In delegated voting, you can delegate your vote on an issue to someone else if you don't have the time or expertise to follow the issue yourself.
One of the pitfalls is that rich entities could literally buy your vote. Most implementations attempt to guard against this one way or another.
See also LiquidDemocracy, which refers to a situation where you can delegate your votes and your delegates can delegate their vote (and yours) on to someone else.
Good question, I'm not sure!
At first glance, I'd say that DelegatedVoting is any system allowing any delegation, whereas LiquidDemocracy is the subcategory of systems allowing sub-delegations ("transitive delegated voting"?). So I think they're not quite the same.
But I'm not sure about that, so if it was just that, we may as well put them on the same page.
However, LiquidDemocracy is a catchy title. And the idea is a pretty radical one which could probably be done in a number of ways. So I predict that LiquidDemocracy will eventually come to have other, specific connotations (hopefully something like "the LiquidDemocracy movement"). So I think we may as well maintain separate pages in anticipation of that time.
Abd ulRahman Lomax: "Delegated Voting" is really "Proxy voting." It has a long history; nearly all for-profit corporations allow it; and that history does show the pitfalls; the basic one is when an entrenched management suggests proxy delegations to clueless shareholders. I've seen the same problem in non-profits which allowed proxy voting. "Delegable Proxy" would not necessarily solve the problem by itself, unless rule or custom discourages the collection of large numbers of primary proxies from individuals. I think this would happen. The problem is not likely to be significant in organizations that do not collect property but rather merely function as communication and coordination systems. See http://beyondpolitics.org