Sleep plays a pivotal role in the brain development and maturation of neonates, and analyzing electroencephalogram (EEG) features across different sleep stages can provide profound insights into the neurophysiological developmental trajectory. In this study, to systematically investigate the neuroelectrical changes associated with advancing gestational age, 1,100 EEG recordings from neonates aged 36 to 45 weeks, collected at the University of Jena, Germany, were analyzed. Following signal preprocessing and artifact removal, frequency-domain features—including power in the delta, theta, alpha, and beta bands, as well as frequency power ratios—and nonlinear features—including Higuchi and Katz fractal dimensions, sample entropy, Poincaré plot parameters, and Recurrence Quantification Analysis (RQA)—were extracted from the C3-T3 and C4-T4 channels. To assess the normality of data distribution, the Kolmogorov-Smirnov test was employed. Hemispheric asymmetries were evaluated using paired t-tests, while inter-group differences across gestational ages were examined using one-way analysis of variance (ANOVA). Furthermore, to model the developmental trajectory of features with advancing gestational age, linear, second-order polynomial, and exponential regression models were applied, and the root mean square error (RMSE) was computed as a goodness-of-fit metric. Paired t-test results indicated no statistically significant differences (p > 0.05) in feature values between the left and right hemispheres, reflecting electrophysiological symmetry and balanced hemispheric function during this developmental period. In contrast, ANOVA revealed statistically significant differences (p < 0.001) in the extracted features across gestational age groups, underscoring the strong influence of gestational age on the evolution of brain activity patterns. Regression modeling revealed that specific frequency power ratios provided the best fit (lowest RMSE) in characterizing the developmental trends of features with age. These findings underscore the importance of frequency power ratios as sensitive and reliable biomarkers for monitoring brain maturation and evaluating neonatal sleep quality.