What is the difference between ArcSDE and ArcGIS Server?
ArcSDE is a component of ArcGIS for Server. However, it is a distinct server technology for providing geodata services from relational database management systems that pre-dates ArcGIS for Server. For more details see What is ArcSDE?
ArcGIS for Server provides a range of web services beyond simple geodata services: Feature, Map, Geocoding, Geoprocessing, Image, Globe, Web Editing, KML, OGC, etc. For more details see What is ArcGIS Server?
I was looking for a clear explanation of what the differences were and I came across a page for ArcGIS 10.3 that described ArcSDE and how it is included in desktop and it's limitations.
Scroll down to the section called.
It looks like one can install the SQL server express edition (with all it's limitation) but you can only do stuff with it if you have at least the Standard license level.