Coded distributed machine learning mitigates straggler effects and provides privacy protection by introducing redundancy through coded computing. However, the system remains vulnerable to privacy breaches when the number of honest-but-curious nodes surpasses the designed threshold, or when outsider adversaries eavesdrop on sensitive data. To address these limitations, we propose a privacy-preserving and straggler-resilient distributed learning framework, namely, differential privacy-based Lagrange coded computing (DP-LCC). First, we design a three-layer protection strategy against privacy threats and stragglers by retaining labels at the master, obfuscating features via Lagrange interpolation, and injecting calibrated noise into local computations. Second, we theoretically prove that the aggregated gradient is an unbiased estimator with bounded variance, and derive convergence bounds under both Gaussian and Laplace mechanisms, revealing the trade-off between privacy budgets and model utility. Third, we provide a comprehensive analysis of the system's computational complexity, privacy composition, and heterogeneity to verify the framework's efficiency and adaptability in realistic distributed environments. Extensive experiments on four benchmark datasets validate the theoretical results, demonstrating the robustness of DP-LCC against varying system parameters and heterogeneous environments.