You need to make a distinction between traditional active loudspeakers and more modern active loudspeakers with DSP.
The traditional ones utilize an “active crossover” to drive multiple power amplifiers and integrate them into the loudspeaker enclosure. These may have advantages over “passive loudspeakers” in that the amps handle only the part of the spectrum required by their particular drivers and the electronic crossover can be more subtle and efficient than a traditional one.
OTOH, DSP-Active loudspeakers add digital processing to (1) further enhance what an analog electronic crossover can do, (2) provide options for matching the loudspeaker to the room and its position within it and (3) keep everything in the digital domain until the internal power amps. In addition, these same speakers are usually configured with special loading and/or additional drivers to control the speaker’s spatial distribution.
I see only a minor advantage in traditional active loudspeakers and do not regard it as a determinative factor in choosing a particular loudspeaker. OTOH, DSP-enable loudspeakers can be substantively different and superior to comparably sized/priced competitors.