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The development and utilization of wind energy is of great significance to the sustainable development of China’s economy and the realization of the “dual carbon” goal. Under typhoon conditions, the randomness and volatility of wind speed significantly impact the energy efficiency and design of wind turbines. This paper analyzed the changes in wind speed and direction using the BFAST method and Hurst index based on data collected at 10 m, 30 m, 50 m, and 70 m heights from a wind power tower in Yancheng, Jiangsu Province. Furthermore, the paper examined the causes of wind speed and direction changes using wind speed near the typhoon center, distance from the typhoon center to the wind tower, topographic data, and mesoscale system wind direction data. The conclusions drawn are as follows: (i) Using the BEAST method, change points were identified at 10 m, 30 m, 50 m, and 70 m heights, with 5, 5, 6, and 6 change points respectively. The change points at 10 m, 30 m, and 50 m occurred around node 325, while the change time at 70 m was inconsistent with other heights. Hurst index results indicated stronger inconsistency at 70 m altitude compared to other altitudes. (ii) By analyzing the wind direction sequence at 10 m, 30 m, 50 m, and 70 m, it was found that the wind direction changes follow the sequence Southeast (SE)—East (E)—Southeast (SE)—Southwest (SW)—West (W)—Northwest (NW). Notably, the trend of wind direction at 70 m significantly differed from other altitudes during the wind speed strengthening and weakening stages. (iii) Wind speed at 10 m and 70 m altitudes responded differently to the distance from the typhoon center and the wind near the typhoon center. The correlation between wind speed and the distance to the typhoon center was stronger at 10 m than at 70 m. The surface type and the mesoscale system’s wind direction also influenced the wind speed and direction. This study provides methods and theoretical support for analyzing short-term wind speed changes during typhoons, offering reliable support for selecting wind power forecast indicators and designing wind turbines under extreme gale weather conditions.