There are two seemingly identical Spatial Analysis tools:
(1) Surface -> Aspect:
Derives aspect from a raster surface. The aspect identifies the downslope direction of the maximum rate of change in value from each cell to its neighbors.
Aspect can be thought of as the slope direction. The values of the output raster will be the compass direction of the aspect.
(2) Hydrology -> Flow direction:
Creates a raster of flow direction from each cell to its steepest downslope neighbor.
Manual comparison
To test this I created an aspect layer and a flow direction layer, converted the aspect layer to the format of flow direction using Raster Calculator:
64*(("aspect" >= 0) & ("aspect" < 22.5)) +
128*(("aspect" >= 22.5) & ("aspect" <67.5)) +
1*(("aspect" >=67.5) & ("aspect" <112.5)) +
2*(("aspect" >=112.5) & ("aspect" <157.5)) +
4*(("aspect" >=157.5) & ("aspect" <202.5)) +
8*(("aspect" >=202.5) & ("aspect" <247.5)) +
16*(("aspect" >=247.5) & ("aspect" <292.5)) +
32*(("aspect" >=292.5) & ("aspect" <337.5)) +
64*(("aspect" >=337.5) & ("aspect" <=360))
and then compared to the flow direction layer:
("aspect_conv" == "flowdir")
The vast majority of the resulting layer is 1 (true) and I suppose the few areas with 0 is because of flat aspects or nulls.
It therefore appears that the only difference is that "Flow Direction" returns unique values (powers of 2), while aspect returns a continuous range of floats. Is that all?