If the impulse responses of the two filters are chosen as sine and a cosine, the only mathematical difference between QAM and CAP waveforms is that the phase of the carrier is reset at the beginning of each symbol.
[1] If the carrier frequency and symbol rates are similar, the main advantage of CAP over QAM is simpler implementation.
[1] The modulation of the baseband signal with the quadrature carriers is not necessary with CAP, because it is part of the transmit pulse.
[1] CAP finds application in HDSL and in early proprietary ADSL variants.
The standardized variants of ADSL, ANSI T1.413 Issue 2 and G.dmt, as well as the successors ADSL2, ADSL2+, VDSL2, and G.fast, do not specify CAP, but rather discrete multi-tone (DMT) modulation.