By 1802, when this map was drawn, there were several dams just above the tidal section of the river. If you enlarge the map, you can see grist mills, linseed oil mills, and a fulling mill. The early industry in Exeter is all dependent on water power and the dams.
In the 1820s, a company formed and bought all the upstream dams and water rights, and built one large dam just downstream of the main bridge. In 1829, the Exeter Manufacturing Company built it's first buildings and started using the power provided by the four water wheels at the dam. It was the only power source available until the 1870s.
I couldn't find any history of how they moved away from water power, but I suspect it was first to steam and then to electricity. If you are familiar with how water power is transmitted to pulleys and shafts, tying that to steam seems a logical step. There was still use for water and the last dam was built in 1914.
Here's Exeter as it was in the 1940s. The dam was already an artifact. Exeter at that time was a mill town with several industries, primary the cotton mill and a shoe factory. The mill owner had most of the political power in town and also a controlling interest in the local bank. It was not an idyllic situation. The hours were long, working conditions poor and dangerous, and it was only changes in the laws that lead to eight hours shifts and improvements in conditions.
It did provide jobs and manufacturing for 150 years.
The Exeter Manufacturing Company was sold in 1966, then sold again in 1981, finally closing in 1983. The mill building was turned into apartments. Somewhere along the way the dam became the responsibility of the town. It served no function. It had been neglected for decades.
A survey found that significant upgrades would be necessary if it was to be maintained and re-certified as safe. It had been known since colonial times that the dam blocked migratory fish from reaching the spawning grounds in the upper fresh water reaches of the river. Fish ladders had been installed in the 1950s without much success. You can see how this was going to go.
No necessarily a bad decision and one that made economic sense as removing the dam was the least expensive option. It was put to a vote and the decision was 2 to 1 in favor. The Exeter River was reopened for the first time since 1638. Here's a documentary. It's not the loss of the unused dam. It's the changes to the town and the people I was noticing.