Should I ever pick the continuous wavelet transform over the discrete one? DWT vs CWT vs STFT
Standard 2-band discrete wavelet transforms have some subsampling, thus they tend to be much less precise in localization. Plus, their shapes are limited, and the ones with finite support are slightly dissymmetric (all but Haar), which is problematic for symmetric pulse/edge detection.

If you don't care about computations, and are interested in visualization, matching, feature detection, CWT might be easier.
If you are interested in  denoising, compression, restoration, DWT are often more appropriate.

Yet, there are many useful schemes bridging the gap between DWT and CWT that can be used in dedicated applications.


A reference paper: Paul S. Addison, 2005, Physiol. Meas. Wavelet transforms and the ECG: a review
References for fast CWT implementations
Matlab code: continuous wavelet transform and inverse
C code: CWT Library: Continuous wavelet transform library
C++ code: Wavelet - A library for online estimation of the Continuous Wavelet Transform