Model updates in edge intelligence (EI) rely on server station centralized processing, suffering from limited uplink bandwidth, prolonged update cycles, underutilized edge computing resources, and catastrophic forgetting. To overcome these challenges, we propose FedCog, a communication-efficient federated continual learning (FCL) framework using task-driven client collaboration and contribution-aware aggregation. FedCog enables collaborative processing of new task data across edge nodes, effectively mitigating catastrophic forgetting through three innovations: Dynamic communication partnership where clients calculate data change rates to skip 70% of communication rounds and dynamically match partners via task relevance scoring; Selective parameter distribution that filters knowledge transfer using task correlation scores, maximizing relevant sharing while blocking interference; and Contribution-oriented gradient aggregation weighting updates by directional alignment with global objectives to accelerate convergence. Building on these strategies, we formulate an optimization problem to minimize communication cost while ignoring rate constraints and accuracy requirements. Extensive experimental results demonstrate that FedCog outperforms state-of-the-art approaches, achieving up to 10.0% absolute accuracy gain and 30.8% communication cost reduction.