Pavement is the material itself, such as asphalt, concrete, etc. The word "pave" means to cover a surface. Sidewalk, road, driveway, and such refer to the use of the paved area. You don't typically call your house "bricks" or "wood" or whatever it may be constructed from, so why call a sidewalk "pavement".
In British English, the terms "pavement" and "paved" also tend to have a strong sense of a surface being covered with "paving stones" as opposed to being covered by a continuously poured material such as tarmac, asphalt or concrete, a process known as "metalling". So the pavements are paved, and the roadway is metalled.
We adopted metalling for the roadway long before using it on the pavements, and it is still very common to find pavements that are flagged (i.e., covered with big flagstones) .
But we also use "footpath" and sometimes "causeway" (pronounced kz.i in some areas) for the bit reserved for pedestrians.