Most edge-cloud collaboration frameworks rely on the substantial computational and storage capabilities of cloud-based artificial neural networks (ANNs). However, this reliance results in significant communication overhead between edge devices and the cloud and high computational energy consumption, especially when applied to resource-constrained edge devices. To address these challenges, we propose ECC-SNN, a novel edge-cloud collaboration framework incorporating energy-efficient spiking neural networks (SNNs) to offload more computational workload from the cloud to the edge, thereby improving cost-effectiveness and reducing reliance on the cloud. ECC-SNN employs a joint training approach that integrates ANN and SNN models, enabling edge devices to leverage knowledge from cloud models for enhanced performance while reducing energy consumption and processing latency. Furthermore, ECC-SNN features an on-device incremental learning algorithm that enables edge models to continuously adapt to dynamic environments, reducing the communication overhead and resource consumption associated with frequent cloud update requests. Extensive experimental results on four datasets demonstrate that ECC-SNN improves accuracy by 4.15%, reduces average energy consumption by 79.4%, and lowers average processing latency by 39.1%.