A weir isn't so much a dam as the overflow from a dam. A couple of examples would be :-
1) A watermill has a dam to hold back the water of a stream, so it can give a "head" to the wheel which powers the mill. Once the dam has filled with water, it needs to escape from the system, A weir was built, usually returning surplus water to the original course of the stream or river. With the industrial revolution, in many places (particularly in the North of England) woollen mills and cotton mills were powered by water turbines. These needed more than just a small stream, so a weir was built across the whole width of a decent sized river, giving a powerful force of water through a small inlet to the turbines.
2) Many of England's canals were built in a course which parallels a river. Some were "Navigations" which means that locks were built in a river, to "flatten" out the downward flow of the water. At a lock, there was the need to allow excess water to return to the next level of the now canalised river. If it flowed over the lock, it could prevent the lock being used. The solution was to divide the river into two channels. One goes to the lock, the other to a weir, which water flows over to the next level.

Back to your songs / stories.
At Bath, if the river is flowing slowly, the water running over the weir will not have had much force behind it, so floating debris such as branches (and corpses) will be caught by the lip of the weir. At high flow - they wash over.
In the Sally Gardens, you refer to grass growing on the weir.
In one sense a grass covered weir is a sign of dereliction. If it was built power a mill, and was still functioning, it would not be covered with grass, the water would wash any seedlings away. If, however, the mill had closed down, the water turbines would have been dismantled and part ot the weir might have been dismantled or broken away, so water runs through the gap left, and does not go over the weir. Hence it becomes covered with grass.
In the song, (originally a poem by William Butler Yates - I believe) the grass on the weir is more likely to be growing on the edge of a weir on a slow flowing river - hence taking life easy. (strange people, these poets...).
Hope this makes our use of weirs in literature a bit more obvious.

Joe - weirs are commonly used over here by the nutters who paddle canoes. They use them on rivers which do not have white water rapids - probably don't have rapids because somebody dammed the river and built a weir - ironic eh!

Quack!!
Geoff the (aquatic correspondent) Duck.