At its most simple:
A dump is an excavated piece of land where people threw trash. They are not longer legal due to the lack of environmental oversight and protection.
A Landfill is a highly engineered excavated pieces of land operated to provide a human health service. They exist to manage waste but are built in a way that keeps environmental health at the forefront by addressing concerns such as groundwater protection and environmental protections.
A more detailed explanation is available at: Differences Between a Dump and Landfill (PDF).
Differences Between a Dump and Landfill (PDF).