For me, a British person, build is associated with everyday items such as houses, roads, bridges, office buildings etc.
The word construct, to me suggests a requirement for an element of ingenuity. They are going to construct an underground reservoir/ a space station/ a hydro-electric dam that utilises its own electricity etc.
If it is bulk standard, it is built, if it is something new and ingenious it is constructed. That is my basic sense of the two words. Though it is a general comment which I accept is not true 100% of the time.
I do not believe we use construct nearly as often as do Americans. Hardly anyone would say, for example, they are going to construct a house on the site. For such everyday things it would always be build, building site etc.
As regards building and construction, if it is occupied by people (e.g. house, office block, department store, factory) it will be a building, if it is a device that is either ornamental (e.g. The Angel of the North) or useful (e.g. a gantry crane for unloading vehicles) it is a construction.