I'm British and, to my mind, a "building company" would be concerned mainly with building houses or small, conventionally built commercial premises. They might also undertake repair, alteration and extention work on existing houses and small commercial premises. The skill set of their employees might include groundwork, brick and block work, joinery, roofing, wiring and pipework.
A "construction company" would be more likely to be concerned with larger buildings (including what we call "blocks of flats" and Americans call "apartment blocks") but also with steel frame and large scale cast concrete commercial building. The skill set of their employees would include those mentioned above but also include structural steel work and, possibly, shuttering for concrete.
There is a complication, however, in that a large "building company" by the above definition which undertook developments consisting of many houses might well be described as a "construction company" even though it had no in-house steel erectors and only built on a domectic scale.