Great answers above, but just wanted to add my voice to the choir :)
ArcObjects
- has some (specialized) functionality that the ArcGIS Runtime SDKs don't provide: complex features, geometric networks, topology, annotation classes, etc.
- is designed for desktop deployments
- has a LARGE footprint (file size and memory)
- is based on COM technology and is restricted to .NET and Java dev enviroments
- is a large investment
ArcGIS Runtime SDK
- performs extremely well
- has a relatively light footprint
- provides free development and testing (see developers.arcgis.com/arcgis-runtime/ for details)
- runs natively on several platforms (Andriod, iOS, Java, .NET, Qt, MacOS, Xamarin is coming soon)
- is designed to run on a variety of computers and devices
IMHO, if you are not doing something specialized that requires ArcObjects, you should use one of the ArcGIS Runtime SDKs.
Some updates:
- ArcGIS Runtime SDK for .NET now supports 3D. The other platforms will in the next release.
- For offline workflows, local data can be read in a few formats (map and tile packages, runtime geodatabase, shapefile)
Full disclosure: I'm a product engineer for the ArcGIS Runtime SDK for .NET, so I'm a little biased!