In this study, we present a multimodal framework for predicting neuro-facial disorders by capturing both vocal and facial cues. We hypothesize that explicitly disentangling shared and modality-specific representations within multimodal foundation model embeddings can enhance clinical interpretability and generalization. To validate this hypothesis, we propose DIVINE (DIsentangled Variational INformation NEtwork), a fully disentangled multimodal framework that operates on representations extracted from state-of-the-art (SOTA) audio and video foundation models, incorporating hierarchical variational bottlenecks, sparse gated fusion, and learnable symptom tokens. DIVINE operates in a multitask learning setup to jointly predict diagnostic categories (Healthy Control, ALS, Stroke) and severity levels (Mild, Moderate, Severe). The model is trained using synchronized audio and video inputs and evaluated on the Toronto NeuroFace dataset under full (audio-video) as well as single-modality (audio-only and video-only) test conditions. Our proposed approach achieves SOTA results, with the DeepSeek-VL2 and TRILLsson combination reaching 98.26% accuracy and 97.51% F1-score. Under modality-constrained scenarios, the framework performs well, showing strong generalization when tested with video-only or audio-only inputs. It consistently yields superior performance compared to unimodal models and baseline fusion techniques. To the best of our knowledge, this is the first framework that combines cross-modal disentanglement, adaptive fusion, and multitask learning to comprehensively assess neurological disorders using synchronized speech and facial video. Code and model weights will be released upon the completion of the double-blind review process.