1. Model Structure: Stan is a probabilistic programming language that allows for the encoding of complex Bayesian models, while XGBoost is a scalable machine learning library specifically designed for gradient boosting. Stan focuses on inference and uncertainty quantification, providing a higher level of flexibility in modeling, whereas XGBoost is more oriented towards optimizing prediction accuracy using an ensemble of decision trees.
2. Model Usage: Stan is best suited for small to medium-sized datasets with complex hierarchical structures and where uncertainty estimation is crucial, making it preferable for academic research and scientific applications. In contrast, XGBoost excels in handling large datasets with high-dimensional features, making it a popular choice in industry settings, especially for competitions like Kaggle.
3. Output Interpretation: Stan provides rich posterior distributions of parameters, enabling users to perform Bayesian inference and make probabilistic predictions. On the other hand, XGBoost generates point predictions based on the ensemble of decision trees, which are valuable for optimizing specific evaluation metrics like accuracy or AUC.
4. Model Training: Stan employs a variety of sampling techniques such as Hamiltonian Monte Carlo and No-U-Turn Sampler for posterior inference, requiring longer computation times but providing accurate estimates, while XGBoost uses a gradient boosting framework to iteratively improve the model by minimizing a differentiable loss function, resulting in faster training times with decent predictive performance.
5. Model Interpretability: Stan offers transparent model specifications with clear priors, allowing users to understand the assumptions and potential biases in the model, while XGBoost offers feature importance scores derived from the ensemble of decision trees, providing insights into the contribution of each input variable to the predictions.
6. Scalability: Stan's computational complexity increases significantly with the number of parameters and data points, limiting its scalability for very large datasets, whereas XGBoost is highly scalable and efficient in handling big data due to its parallel and distributed computing implementations.

In Summary, Stan and XGBoost differ in their modeling approach (Bayesian vs. ensemble-based), usage scenarios, output interpretation, training methodology, interpretability, and scalability, catering to different needs in research and industry applications.