No, a clear differentiation exists in their areas of expertise.
Plumbers specialize in central heating and hot and cold water pipes, etc. Their everyday tasks involve installing or repairing boilers, radiators, pipes, pumps, showers, basins, and toilets.
A bathroom fitter performs a wide variety of activities. They may need to acquire skills in various trades, such as plumbing, tiling, joinery, electrical work, construction, and drywall hanging.