A mower is a person or machine that cuts (mows) grass or other plants that grow on the ground. Usually mowing is distinguished from reaping, which uses similar implements, but is the traditional term for harvesting grain crops, e.g. with reapers and combines.
Eicher tractor with a mid-mounted finger-bar mower
Rotary cutters mounted on a swather
Reel mower
Flail mower
A reaper is a farm implement or person that reaps crops at harvest when they are ripe. Usually the crop involved is a cereal grass. The first documented reaping machines were Gallic reapers that were used in Roman times in what would become modern-day France. The Gallic reaper involved a comb which collected the heads, with an operator knocking the grain into a box for later threshing.
Typical 20th-century reaper, a tractor-drawn Fahr machine
A reaper cutting rye in Germany in 1949
1900 ad for McCormick farm machines--your boy can operate them
McCormick's reaper at a presentation in Virginia