[d]irect democracy (also known as pure democracy) is a form of democracy in which people decide (e.g. vote on, form consensus on) policy initiatives directly. (...) [D]irect democracy might entail passing executive decisions, the use of sortition, making laws, directly electing or dismissing officials, and conducting trials. Two leading forms of direct democracy are participatory democracy and deliberative democracy.
This differs from representative democracy where people elect their representatives for a certain amount of time and these representatives can do "whatever they see fit" (in the limitations defined by the law) for the benefit of their electors. Usual complaints about representative democracy is that representatives don't always follow the program they proposed before their election.
On the other hand, anarchy is defined by Wikipedia as
a political philosophy that advocates self-governed societies with voluntary institutions. These are often described as stateless societies, but several authors have defined them more specifically as institutions based on non-hierarchical free associations.
These systems seem very similar to me. To be very schematical, in both systems, people sit down and discuss, and decide altogether on the way to conduct the society.
Hence my question: what are the differences in principles and practice between the two systems?