Given the application of a multiple regression and artificial neural networks (ANNs), this paper describes development of models for predicting surface roughness, linking an arithmetic mean deviation of a surface roughness to a torque as an input variable, in the process of drilling enhancement steel EN 42CrMo4, thermally treated to the hardness level of 28 HRC, using cruciform blade twist drills made of high speed steel with hardness level of 64–68 HRC. The model was developed using process parameters (nominal diameters of twist drills, speed, feed, and angle of installation of work pieces) as input variables varied at three levels by Taguchi design of experiment and measured experimental data for a torque and arithmetic mean deviation of a surface roughness for different values of flank wear of twist drills. The comparative analysis of the models results and the experimental data, acquired for the inputs at the moment when a wear span reaches a limit value corresponding to a moment of the drills blunting, demonstrates that the neural network model gives better results than the results obtained in the application of multiple linear and nonlinear regression models.