why is there a Darcy friction factor and a Fanno friction factor that are different
The friction factor is the Fanning friction factor. There is also a concept known as Fanno flow, and that is related to viscous stresses at walls too, but that’s different. Fanno was Italian. Fanning was… probably American or British. I bet Darcy was French. This information is not useful.
The Darcy and Fanning friction factors come from two almost identical arguments that are just different by a factor of 4.
If you are thinking about head loss in a pipe, you are working with the Bernoulli constant (or equivalently a total pressure). If you nondimensionalize the Bernoulli constant (or the Bernoulli constant losses) by the dynamic pressure, you end up with what is typically called the Darcy friction factor, which I call the viscous dissipation factor:
f_D = \frac{\Delta B}{\frac12\rho U^2}\frac{d}{L}
alternately, you could, instead of thinking about the macroscopic parameter B, be thinking microscopically about the shear stress at the wall \tau_{rz}, and you could normalize that by the dynamic pressure:
f_F = \frac{\tau_{rz}}{\frac12\rho U^2}
both of these are perfectly sensible ways to nondimensionalize properties related to viscous dissipation in pipe flow. Head loss is proportional to dynamic pressure. Wall shear stress is proportional to dynamic pressure. However, the two factors are not equal.
Take a cylindrical section of flow in a pipe of length dz. The net pressure force on that section is -\frac{\partial B}{\partial z} dz times the cross-sectional area \pi d^2/4. The net viscous stress a the wall is \tau_{rz} times the wall surface area \pi d\, dz. At steady state these sum to zero:
-\frac{\partial B}{\partial z} dz\left(\frac{\pi d^2}{4}\right)+ \tau_{rz}\pi d\, dz=0
redefining in terms of the friction factors:
-\frac{f_D}{d}\left(\frac12\rho U^2\right) dz\left(\frac{\pi d^2}{4}\right)+ \left(\frac12\rho U^2\right)f_F\pi d\, dz=0
leading to
4f_F = f_D
or we could have rearranged the original equation to get
\tau_{rz} =\frac{\partial B}{\partial z} \left(\frac{d}{4}\right)
both of these equations show that the magnitude of the stress and the magnitude of the bernoulli constant drop are related but different by a factor of 4.