The central difference is that in a direct democracy, the people make decisions themselves, whereas in a representative democracy, representatives make decisions for the people.
Direct democracy is seen as a purer form of democracy. Democracy concerns enforcing the will of the people. When the people make decisions themselves, there will is automatically enforced. Representative democracy is a less purer form. Representatives must weigh the will of the people against practical considerations, party policies, political ideology and electoral interests. As a result, the will of the people will always be moulded around these considerations, and the ultimate decision will rarely, if ever, mirror the exact will of the people. At least in comparison to direct democracy.
In modern, 21st century democracies, direct democracy is often used in exceptional circumstances to decide matter of political controversy, where the issues involved are relatively simple. In comparison, representative democracy is often the rule, rather than the exception, used to solve matter of political complexity, where the issues involved are diverse. The nature of the problems each form of democracy is used to resolve are often very different.