There is a recent trend towards integrating resting state functional magnetic resonance imaging (RS-fMRI) and diffusion MRI (dMRI) for brain connectivity estimation, as motivated by how estimates from these modalities are presumably two views reflecting the same underlying brain circuitry. In this paper, we show on a cohort of 60 subjects that conventional functional connectivity (FC) estimates based on Pearson's correlation and anatomical connectivity (AC) estimates based on fiber counts are actually not that highly correlated for typical RS-fMRI (~7 min) and dMRI (~32 gradient directions) data. The FC-AC correlation can be significantly increased by considering sparse partial correlation and modeling fiber endpoint uncertainty, but the resulting FC-AC correlation is still rather low in absolute terms. We further exemplify the inconsistencies between FC and AC estimates by integrating them as priors into activation detection and demonstrating significant differences in their detection sensitivity. Importantly, we illustrate that these inconsistencies can be useful in fMRI-dMRI integration for improving brain connectivity estimation.