A couple of similar and related questions which did not prove useful to me are:
The best is to check ESRIs product information sites on its products, they describe its key features pretty well.
In short words:
ArcGIS Engine is a pure Developer Toolkit to embed ESRIs GIS Technology (Map View controls and so on) in individual Desktop and/or Mobile (Windows 6.5 and 7 mobile) Applications which are not intended as a pure GIS application at first place, more as an "Desktop application with map visualization capabilities" . To run such an application, you only need an Engine Runtime License which is the cheapest option ESRI provides
For sure you also can create a Desktop-like or even mobile App with ArcGIS Server frameworks, but the content is always served from ArcGIS Server over the internet (while with Engine you completely embed data, functionality and visualization into your target device).