A Weir is intended to have the water flow over the whole of the top surface on a fairly regular basis - most of the ones in the country area around Bundaberg where I grew up often had a (concrete not asphalt) road surface on the top - wide enough to cross the river in dryish times. It forms a buffer, retaining some water for drier times in a pond area upstream. None of the weirs in Australia were ever likely to have been associated with any sort of mill construction, such things were things of history (power was mostly generated by steam or ICEs) when these Australian ones were constructed - they were mostly used for irrigation or for a supply of potable water in drier times. Most rivers in Eastern Australia are short and fast flowing during times of rain, and lower and slower between wet periods. Weirs are not uncommon on many smaller streams in country areas.

A large Dam is intended to hold water back and deal with the flow through a regulated outlet (possibly also generating electrical power) - if the water reaches the top enough to start to flow over, it is a serious cause for concern by the engineers, as due to it's design and construction, this may cause serious damage to the dam. There may be a 'spillway' which is intended to deal with small amounts of flow over once the optimum high level is reached, but if considerable stream flow is anticipated, the outputs are opened to stop the dam overtopping.

But small farm dams may rely on the water mostly behaving like a weir - although the volume is much smaller, and towns are not likely to be swept away downstream if they break.

Robin