||A coverage has perfect fabric topology, no gaps and no overlaps. The essense of editing coverages is editing arcs and labels, and when you are done you rebuild the polygons, abutting polygons sharing the same arc and hence same vertices making up that arc.
Shapefiles and feature classes have "independent" features, so that each polygon shape has its own shape field and is linked to an attribute record. Thus they can overlap or have gaps. When they are coincident each polygon shape has a line for their shared boundary. If you edit a polygon, say by splitting it, you've added vertices to its shape, but not those of the abutting shapes, and given the limits of precision, where the polygons previously had coincident lines they no longer do.
Shapefiles and feature classes can be set up in a way that sort of mimics coverages, by using lines and points. You edit the lines and points (including the attributes) and when you are done build polygons using both. The polygons have coincident lines where they abut because they come from the same source.
Shapefiles and feature classes can be edited in ArcMap using map topology tools as a way to reduce the likelihood of gaps or overlaps occurring, but I don't think that helps with issues raised by actions like the cutting of polygons, more with the moving of vertices or reshaping or adding polygons, and it doesn't deal with copying and pasting, merging or appending polygons, although using the construct features tool can be invoked.
The matter of topology rules, however, can go far beyond fabric topology. Thus in a geodatabase, not only can one have rules addressing overlaps or gaps, but there can be all sorts of rules covering other matters such as what kind of attributed lines can connect to other kinds of attributed lines. ESRI materials on topology go into these in depth. These are not available for shapefiles or coverages.
We predominately use coverages because the fabric topology guarantees accurate acreages and so far at least, only coverages can feasibly handle our complex overlays of large datasets. But I also use shapefiles in the lines and points mode to mimic coverages for smaller projects since ArcEdit has its own limitations and difficulties.