In this paper, we propose a novel sparse data-to-symbol neural network (SDSNN) receiver for bandwidth-limited underwater visible light communication (UVLC) based on non-orthogonal multi-band carrierless amplitude and phase modulation (NM-CAP). Bandwidth limited NM-CAP signals usually carry severe inter-symbol interference (ISI) and inter-band interference (IBI). The SDSNN receiver directly converts the received NM-CAP data with ISI and IBI into quadrature amplitude modulation symbols without distortion for each sub-band. In contrast, the conventional receiver requires the least mean square (LMS) equalizer to cancel ISI, and the subcarrier component extraction with complex independent component analysis (SCE-ICA) to cancel IBI, respectively. SDSNN provides a novel receiving structure to replace post-equalization, matched filtering, and SCE-ICA. A blue-LED based UVLC system has been demonstrated utilizing NM-CAP16 with 3 sub-bands. The experimental results show that NM-CAP with the SDSNN receiver case reaches the highest spectral efficiency, where an enhancement of 43%, 20%, 6% has been measured over the orthogonal multi-band CAP case, NM-CAP with LMS equalizer case, and NM-CAP with joint LMS equalizer and SCE-ICA case, respectively. Compared with joint LMS equalizer and SCE-ICA case, the proposed SDSNN receiver can achieve 98% reduction of computational complexity.