The increasing penetration of renewable energy sources has introduced significant variability into modern power systems, making accurate short-term forecasting essential for reliable planning and operation. Among renewable resources, wind energy is particularly challenging due to its strong intermittency and nonlinear temporal behavior. This study presents a comparative evaluation of classical statistical and deep learning models for short-term wind speed forecasting, including Moving Average (MA), Simple Exponential Smoothing (SES), Double Exponential Smoothing (DES), Triple Exponential Smoothing (TES), Autoregressive Integrated Moving Average (ARIMA), and Long Short-Term Memory (LSTM). The models are assessed using a real-world wind speed dataset from California covering the years 2021--2022 at a 5-minute resolution. To ensure a fair comparison, model hyperparameters are tuned using a rolling-window validation framework, after which final performance is reported on a 30\% hold-out test set. In addition, robustness is investigated under different additive white Gaussian noise (AWGN) levels. The results show that LSTM achieves the best overall forecasting accuracy, while ARIMA remains highly competitive and robust under noisy conditions, which shows the importance of modeling temporal dependence and noise resilience for reliable short-term wind speed forecasting.