Digital EQ is wonderful and there is a huge amount of hype and marketing for auto-EQ equipment (hook up a microphone and press a button and presto your sound is "optimized") but it is important to recognize the limitations. First and foremost it is always better to improve room acoustics, speaker and listening positions. After as much acceptable acoustic improvements have been made then it falls to EQ (with digital EQ being the most flexible). Below 100 Hz the wavelengths are 10 feet or longer. These frequencies are a reasonable target for notch filtering as you can remove nasty room modal resonant peaks and have an effect on nearly the entire room because the wavelengths are long enough.
Above 100 Hz my recommendation is not to use any sharp EQ. I am fully on board with any broad treble of bass tilt adjustments to suit your tastes and room general issues but nothing else (for example a room with a tile floor will probably be too bright unless you turn down the treble a wee bit)! The issue is that at 1000 Hz you have wavelengths of the order of 1 foot - this means that you have a peak and a trough of a sound wave over a distance of a mere 6 inches. This makes it impractical to have any meaningful effect on the room at all (the benefit may be fine for one microphone position but detrimental for another). All you can do is correct a little for problems with the recording itself or the equipment itself (and, as with the room, it is first and foremost better to get equipment that works for you without needing tweaks)
In any case room EQ is an important topic but a separate issue from active vs passive speakers as it can be applied to both.
As Doug and several others have mentioned - ridding yourself of the passive X-over and the limitations of passive filtering of high power signals is perhaps the biggest advantage of "active". For the same reasons that single driver speakers sound so lively and clear (but have limited frequency response), active speakers (or at least well designed ones) all share a lively dynamic sound as if a veil has been removed. I believe this is because of the removal of the crossover. Passive crossovers rob some life from the music over quite a wide region - up to a couple of octaves. These passive crossover issues have manifested themselves in the need for massive monoblock amplifiers to breathe life back into the music. (If we look at Active speakers then we find the amplifiers are rather modest in size for the equivalent power output)