Background and Objectives: To our knowledge, this is the first study that investigated the prognostic value of radiomics features extracted from not only staging 18F-fluorodeoxyglucose positron emission tomography (FDG PET/CT) images, but also post-induction chemotherapy (ICT) PET/CT images. This study aimed to construct a training model based on radiomics features obtained from PET/CT in a cohort of patients with locally advanced head and neck squamous cell carcinoma treated with ICT, to predict locoregional recurrence, development of distant metastases, and the overall survival, and to extract the most significant radiomics features, which were included in the final model. Materials and Methods: This retrospective study analyzed data of 55 patients. All patients underwent PET/CT at the initial staging and after ICT. Along the classical set of 13 parameters, the original 52 parameters were extracted from each PET/CT study and an additional 52 parameters were generated as a difference between radiomics parameters before and after the ICT. Five machine learning algorithms were tested. Results: The Random Forest algorithm demonstrated the best performance (R2 0.963–0.998) in the majority of datasets. The strongest correlation in the classical dataset was between the time to disease progression and time to death (r = 0.89). Another strong correlation (r ≥ 0.8) was between higher-order texture indices GLRLM_GLNU, GLRLM_SZLGE, and GLRLM_ZLNU and standard PET parameters MTV, TLG, and SUVmax. Patients with a higher numerical expression of GLCM_ContrastVariance, extracted from the delta dataset, had a longer survival and longer time until progression (p = 0.001). Good correlations were observed between Discretized_SUVstd or Discretized_SUVSkewness and time until progression (p = 0.007). Conclusions: Radiomics features extracted from the delta dataset produced the most robust data. Most of the parameters had a positive impact on the prediction of the overall survival and the time until progression. The strongest single parameter was GLCM_ContrastVariance. Discretized_SUVstd or Discretized_SUVSkewness demonstrated a strong correlation with the time until progression.