Spiking Neural Networks (SNNs) are seen as an energy-efficient alternative to traditional Artificial Neural Networks (ANNs), but the performance gap remains a challenge. While this gap is narrowing through ANN-to-SNN conversion, substantial computational resources are still needed, and the energy efficiency of converted SNNs cannot be ensured. To address this, we present a unified training-free conversion framework that significantly enhances both the performance and efficiency of converted SNNs. Inspired by the biological nervous system, we propose a novel Adaptive-Firing Neuron Model (AdaFire), which dynamically adjusts firing patterns across different layers to substantially reduce the Unevenness Error - the primary source of error of converted SNNs within limited inference timesteps. We further introduce two efficiency-enhancing techniques: the Sensitivity Spike Compression (SSC) technique for reducing spike operations, and the Input-aware Adaptive Timesteps (IAT) technique for decreasing latency. These methods collectively enable our approach to achieve state-of-the-art performance while delivering significant energy savings of up to 70.1%, 60.3%, and 43.1% on CIFAR-10, CIFAR-100, and ImageNet datasets, respectively. Extensive experiments across 2D, 3D, event-driven classification tasks, object detection, and segmentation tasks, demonstrate the effectiveness of our method in various domains. The code is available at: https://github.com/bic-L/burst-ann2snn.