To calculate the friction factor we used to use Colebrook equation for fully developed turbulent pipe flow. Also, a graphical method is created by Lewis Moody using Colebrook equation. However this method had some disadvantages that Churchill tried to address:
Colebrook equation is implicit in 'f' that is the friction factor appears in both sides of the equation. Whereas Churchill equation is explicit equation.
Colebrook equation is applicable only to turbulent pipe flow. Whereas Churchill equation can be used for any Reynolds number and for any roughness as well as for transition region.