This is the third in a series of three papers exploring the connection between the multiwavelength properties of AGN in nearby early-type galaxies and the characteristics of their hosts. Starting from an initial sample of 332 galaxies, we selected 116 AGN candidates requiring the detection of a radio source with a flux limit of ∼1mJy, as measured from 5GHz VLA observations. In Capetti and Balmaverde (2005A&A...440...73C
, Paper I) we classified the objects with available archival HST images into ``core'' and ``power-law'' galaxies, discriminating on the basis of the nuclear slope of their brightness profiles. We used HST and Chandra data to isolate the nuclear emission of these galaxies in the optical and X-ray bands, thus enabling us (once combined with the radio data) to study the multiwavelength behaviour of their nuclei. The properties of the nuclei hosted by the 29 core galaxies were presented in Balmaverde and Capetti (2006A&A...447...97B
, Paper II). Core galaxies invariably host a radio-loud nucleus, with a median radio-loudness of Log R=3.6 and an X-ray based radio-loudness parameter of Log RX
=-1.3. Here we discuss the properties of the nuclei of the 22 ``power-law'' galaxies. They show a substantial excess of optical and X-ray emission with respect to core galaxies at the same level of radio luminosity. Conversely, their radio-loudness parameters, Log R∼1.6 and Log RX
~-3.3, are similar to those measured in Seyfert galaxies. Thus the radio-loudness of AGN hosted by early-type galaxies appears to be univocally related to the host's brightness profile: radio-loud AGN are only hosted by core galaxies, while radio-quiet AGN are found only in power-law galaxies. The brightness profile is determined by the galaxy's evolution, through its merger history; our results suggest that the same process sets the AGN flavour. In this scenario, the black holes hosted by the merging galaxies rapidly sink toward the centre of the newly formed object, setting its nuclear configuration, described by e.g. the total mass, spin, mass ratio, or separation of the SMBHs. These parameters are most likely at the origin of the different levels of the AGN radio-loudness. This connection might open a new path toward understanding the origin of the radio-loud/radio-quiet AGN dichotomy and provide us with a further tool for exploring the co-evolution of galaxies and supermassive black holes.