The clinical application of cone-beam computed tomography (CBCT) is constrained by the inherent trade-off between radiation exposure and image quality. Ultra-sparse angular sampling, employed to reduce dose, introduces severe undersampling artifacts and inter-slice inconsistencies, compromising diagnostic reliability. Existing reconstruction methods often struggle to balance angular continuity with spatial detail fidelity. To address these challenges, we propose a Continuity-driven Synergistic Diffusion with Neural priors (CSDN) for ultra-sparse-view CBCT reconstruction. Neural priors are introduced as a structural foundation to encode a continuous threedimensional attenuation representation, enabling the synthesis of physically consistent dense projections from ultra-sparse measurements. Building upon this neural-prior-based initialization, a synergistic diffusion strategy is developed, consisting of two collaborative refinement paths: a Sinogram Refinement Diffusion (Sino-RD) process that restores angular continuity and a Digital Radiography Refinement Diffusion (DR-RD) process that enforces inter-slice consistency from the projection image perspective. The outputs of the two diffusion paths are adaptively fused by the Dual-Projection Reconstruction Fusion (DPRF) module to achieve coherent volumetric reconstruction. Extensive experiments demonstrate that the proposed CSDN effectively suppresses artifacts and recovers fine textures under ultra-sparse-view conditions, outperforming existing state-of-the-art techniques.