High frequencies disappear when applying discrete wavelet transform
The wavelet transform is usually constructed in such a way that in the reconstruction the input signals are either convoluted with each of the downsampled filters and then summed up, or equivalently, are upsampled filling the gaps with zeros.

Upsampling with interpolation is equal to applying an additional filter $[0.5,1,0.5]$ which is a low-pass, smoothening filter. Esp. on the high-frequency or detail component this will lead to large errors.