This paper investigates whether structural econometric models can rival machine learning in forecasting energy--macro dynamics while retaining causal interpretability. Using monthly data from 1999 to 2025, we develop a unified framework that integrates Time-Varying Parameter Structural VARs (TVP-SVAR) with advanced dependence structures, including DCC-GARCH, t-copulas, and mixed Clayton--Frank--Gumbel copulas. These models are empirically evaluated against leading machine learning techniques Gaussian Process Regression (GPR), Artificial Neural Networks, Random Forests, and Support Vector Regression across seven macro-financial and energy variables, with Brent crude oil as the central asset. The findings reveal three major insights. First, TVP-SVAR consistently outperforms standard VAR models, confirming structural instability in energy transmission channels. Second, copula-based extensions capture non-linear and tail dependence more effectively than symmetric DCC models, particularly during periods of macroeconomic stress. Third, despite their methodological differences, copula-enhanced econometric models and GPR achieve statistically equivalent predictive accuracy (t-test p = 0.8444). However, only the econometric approach provides interpretable impulse responses, regime shifts, and tail-risk diagnostics. We conclude that machine learning can replicate predictive performance but cannot substitute the explanatory power of structural econometrics. This synthesis offers a pathway where AI accuracy and economic interpretability jointly inform energy policy and risk management.