What are the pros and cons of wavelet for filtering compared to conventional filters?
Let us first consider the orthogonal DWT: this transform is build with several constraints: orthogonality (and invertibility) of course, and the discretization of some continuous wavelet transform, with a specific dyadic structure, yielding a frequency decomposition whose cut are approximately:

$$[0\;1/2^L\;1/2^{L-1}\ldots 1/2 \;1]$$

Discrete wavelets can be implemented as a bank of filters (or a filterbank) dedicated  to one of the above intervals, followed by the appropriate subsampling. And only some specific filters (albethey in infinite number) obey the above constraints. 

Hence, with the DWT, only dyadic boundaries are available (*), only  a subset of filters can be used, and outputs are decimated. And orthogonality, scales are not used at all.

Whereas with standard filters, you only have linearity constraints: invertibility is not required, you can specify the bounds, ripples and decay you need (at the price of design issues).

So for mere filtering, I cannot see an advantage of using discrete and critical wavelet schemes. However, DWT can do some crude band-pass filtering while being used for other processing needs.

Wavelets now play a more interesting role for data sparsification, along with non-linear analysis, for compression, restoration, which standard filters generally cannot achieve.

As a side note, if you allow redundancy in the transformation, choosing a wavelet with a sufficient number of oscillations in a sufficiently long "window" could do a good job, but that would be overkill.

(*) with $M$-band wavelets and wavelet packets, you could get more generic $M$-ary integers $k/M^n$, but not all fractions or reals