The mesh is the data describing the points that make up the object (This includes faces and edges). This is just one small part of the model.
"Model" is a generic term. It means any data related to drawing a 3d item onto the screen. Exactly what data this includes depends on the program or file format. In some case it might just be the mesh, although usually it also includes some of the following:
Material - the colours, shading models, and other things that determine how to apply lights and rendering
Textures - the images that are mapped onto the mesh
UV - co-ordinates that determine how textures are applied to the mesh
vertex groups - used to break the model into smaller chunks for special purposes
bones / armatures - used to link parts of the mesh to specific parts of animations
animations - how the mesh moves
... the list goes on, depending on what you're trying to do with the mesh.
The difference is what actually gets saved. The 3ds file only saves the model data itself (geometry, materials, etc) while the blender file contains structures explaining the data as well (the so-called DNA structure) that guarantees both forward and backward compatibility. Add to this the workspace setup (tool settings, layouts, preferences, etc), the blender file actually contains far more data.
(To be very specific, the blender file is a direct dump of the internal data structure, while the 3ds is an optimized format to store only very specific data)