Question about the Haar Wavelet transform
The FFT is much better at detecting a sinusoid than a DWT.  The FFT is approximating a periodic signal with a series of periodic signals.  The coefficients of the FFT will be maximum in the frequency bins (could be a single bin with sufficient resolution) where the component of the FFT series best matches the periodic signal to be detected.  The noise will distribute roughly evenly in all bins, so the signal to be detected will show up readily.

The DWT is representing the signal to be detected (in this case a periodic signal) with a set of non-periodic functions, so this requires lots of components (bands as you refer to them – but DWT really works with scale not frequency) to follow the periodic nature of the signal to be detected.  The DWT is better suited to detect / analyze non-periodic signals.