The alluvial plain near a lowland river tends to be very fertile because the soil around it is regularly exchanged. While the "Fertile Crescent" eventually wore out and suffered from desertification due to overfarming, the Nile's flood plain, particularly the delta area, remained fruitful through thousands of years of continuous farming. The primary difference was that the Nile floods regularly, which refreshes the nutrients in the soil. The Tigris and Euphrates flood plains have remained fertile, but not to the same extent because their flooding is less reliable. The rest of the land in that area was merely irrigated from the rivers and did not receive the benefit of the floods. As a result, the nutrients were depleted, salt levels rose, and the desert claimed what was once very good farmland.
More obviously, of course, vegetation tends to be thicker around rivers because that's where the water is! In the American Great Plains (where I grew up), the tall grasses will grow everywhere, but you only naturally find trees near the rivers because the rest of the land just doesn't have enough moisture to support them. Trees planted far from the rivers tend to be rather sickly, and they often fall down during windstorms. Typically on someone's car.