What is GIS?

Geographic Information Systems (GIS) are systems of hardware, software, and procedures to facilitate the management, manipulation, analysis, modelling, representation, and visualization of geo-referenced data to solve complex problems regarding the planning and management of resources.
It is the ability to produce a map image that sets GIS apart from other database systems. Perhaps more simply, a GIS gives you tools to answer questions about a location (geographic information). You can combine layers of information stored in a database that may include addresses, aerial photographs, dates, and other numerical values.

What is the difference between GIS and a traditional map?

Both GIS and traditional maps represent the real world. One way of thinking about it is that the GIS is the Data while the map is the Picture.
The end result of GIS analysis is often a Map, or Cartographic output, but it is only one of many output types. Once created, a traditional map cannot be changed. However, as an integrated system, a GIS allows users to ask new questions of a database and visualize answers.