In prior centuries, a watch was the product of a long period of labor by a single individual that would make, by hand, every component of the timepiece from plates, wheels and springs to the case. In these times, one person would make, from scratch each component. Every completed piece was unique. This is the source of the tradition of "signed" or later called "named" movements.
During the early industrial revolution, especially in England and in Europe otherwise, watchmaking became more of a cottage industry wherein individuals would do specialized portions the work, passing a watch from worker to worker until the watch was complete. Specialized sources developed for complex parts, and for the basic raw "plates" that form the foundation parts of a watch. Frequenly such parts would even be imported from other countries.
Moving watch manufacturing from this stage to a mechanized factory setting, which Americans pioneered in the mid 19th century, was an easy step. In early factories, trays of watches moved from work station to work station, where workers would use bins of machine-made parts to do their portion of the process.
The term "watchmaker" over this time came to refer to the designers of watches, and the skilled crafts people that continued to repair and maintain watches individually. One may ask, isn't the worker in the 19th century factory a watchmaker? Not really, the assembly line workers are really watch assemblers, capable of doing a limited subset of specific tasks, without the need of a detailed understanding of the mechanism.
In modern usage, as then, a watchmaker has a more complete understanding of the history, theory and design of a watch, together with the ability to repair or create from scratch, at least most, if not all, the components of a watch - something the watch assembler does not do.