Electroencephalography (EEG) is highly susceptible to artifact contamination, such as electrooculographic (EOG) and electromyographic (EMG) interference, which severely degrades signal quality and hinders reliable interpretation in applications including neurological diagnosis, brain-computer interfaces (BCIs), etc. Effective EEG denoising remains challenging because different artifact sources exhibit diverse and temporally varying distributions, together with distinct spectral characteristics across frequency bands. To address these issues, we propose BandRouteNet, an adaptive frequency-aware neural network for EEG denoising that jointly exploits band-specific processing and full-band contextual modeling. The proposed model performs band-wise denoising to explicitly capture frequency-dependent artifact patterns. Within this framework, we introduce a routing mechanism that adaptively determines where and to what extent denoising should be applied across temporal locations within each frequency band. In parallel, a full-band conditioner directly processes the original noisy EEG to extract global temporal context, producing both conditional parameters for modulating the band-wise pathway and a coarse-grained signal-level refinement to supplement the final reconstruction. Extensive experiments on the EEGDenoiseNet benchmark dataset demonstrate that BandRouteNet outperforms other methods under EOG, EMG, and mixed-artifact conditions in terms of Relative Root Mean Square Error (RRMSE) and Signal-to-Noise Ratio Improvement (SNR$_{\text{imp}}$) under unified experimental settings, while remaining highly parameter-efficient with only 0.2M trainable parameters. These results highlight its strong potential for high-performance EEG artifact removal in resource-constrained applications.