[ABRIDGED] 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. We selected 116 AGN candidates requiring a radio flux of 1 mJy. We classified the objects with HST images into ``core and ``power-law galaxies, on the basis of the nuclear slope of their brightness profiles. We used HST and Chandra data to isolate their nuclear emission to study the multiwavelength behaviour of their nuclei. The properties of the nuclei hosted by the 29 core galaxies were presented in 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 R,X = -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 R,X ~ -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 hosts 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 galaxys evolution, through its merger history; our results suggest that the same process sets the AGN flavour.
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