The biconical Red Rectangle nebula exhibits very strong unidentified infrared (UIR) emission bands, a subset of the optical diffuse interstellar bands (in emission) and extended red emission (ERE). A key question is the extent to which the carriers of these spectroscopic signatures may be related. In a new study of the 3.3-mu m emission, CGS 4 spectra were recorded at UKIRT, which give information on the spatial distribution of the 3.3-mu m carrier in the nebula and on the width, peak wavelength and profile of the feature as a function of offset from the central star, HD 44179. Both Type I (lambda(0) similar to 3.289 mu m, full width at half-maximum FWHM similar to 0.042 mu m) and Type 2 (lambda(0) similar to 3.296 mu m, FWHM similar to 0.020 mu m) 3.3-mu m features, as defined by Tokunaga et al., are found within the nebula. Type 2 is seen predominantly towards the central star, at the bicone interfaces and east and west of the star in the nebula. The broader Type I feature appears in the nebula 5 arcsec south of the central star, whereas the 3.3-mu m band at 5 arcsec north appears to be a blend of Type 1 and Type 2. We find that there is no significant correlation between the intensity of the 3.3-mu m feature and that of either the unidentified optical (diffuse) emission bands or ERE. This result suggests that there is at most an indirect link between the carrier(s) of the 3.3-mu m band and this subset of diffuse bands. Such a link could arise, for example, if these diffuse band carriers were formed by chemical erosion or photodissociation of polycyclic aromatic hydrocarbon material.