Accurate segmentation of retinal vessels is crucial for the clinical diagnosis of numerous ophthalmic and systemic diseases. However, traditional Convolutional Neural Network (CNN) methods exhibit inherent limitations, struggling to capture long-range dependencies and complex nonlinear relationships. To address the above limitations, an Adaptive Dual Branch Kolmogorov-Arnold UNet (DB-KAUNet) is proposed for retinal vessel segmentation. In DB-KAUNet, we design a Heterogeneous Dual-Branch Encoder (HDBE) that features parallel CNN and Transformer pathways. The HDBE strategically interleaves standard CNN and Transformer blocks with novel KANConv and KAT blocks, enabling the model to form a comprehensive feature representation. To optimize feature processing, we integrate several critical components into the HDBE. First, a Cross-Branch Channel Interaction (CCI) module is embedded to facilitate efficient interaction of channel features between the parallel pathways. Second, an attention-based Spatial Feature Enhancement (SFE) module is employed to enhance spatial features and fuse the outputs from both branches. Building upon the SFE module, an advanced Spatial Feature Enhancement with Geometrically Adaptive Fusion (SFE-GAF) module is subsequently developed. In the SFE-GAF module, adaptive sampling is utilized to focus on true vessel morphology precisely. The adaptive process strengthens salient vascular features while significantly reducing background noise and computational overhead. Extensive experiments on the DRIVE, STARE, and CHASE_DB1 datasets validate that DB-KAUNet achieves leading segmentation performance and demonstrates exceptional robustness.