You're correct that simply putting a dam in place, once its lake is filled, doesn't change the average flow downstream by more than a few percent (those few percent can be lost to increased evaporation). However,
- Many dam systems are used to abstract water from the river for irrigation and other uses, sometimes on a very large scale. This does reduce the average flow downstream.
- Depending on the site, filling the lake might take a year or more, which could have severe consequences on people downstream.
- More generally, dams allow regulation of the downstream flow. This is a major reason for their construction, in areas that want to smooth out a flood/drought cycle. But it means that downstream water supply is then under the control of the dam operators, which may be objectionable to people downstream - especially when the river crosses international borders. The water must be released eventually, but the upstream operator could choose to (for example) withhold it for the growing season and cause crop failure that year.
For all of these reasons, river management is a common subject of international treaties. I don't know the details of this particular case involving Ethiopia and Egypt, but I imagine that the main concern is probably about water abstraction. If Ethiopia takes water for drinking, irrigation, etc., then that's less water left for Egypt.