Is there a difference between active and powered speakers?
Yes! Steve beat me to it.
In design, a powered speaker schematically is identical to a passive speaker: signal goes to each channel of a power amp; then to a crossover; then to each driver. The advantages of the powered speaker are: 1.) Save money on buying a cabinet; 2.) Manufacturer has chosen a well matched amp for the given speaker (but of course that takes away the consumer's option to pick their own power amp and show it off). And if you're a tube power amp fan, you're all but stuck with passive speakers.
An active speaker schematically is different: signal goes to the crossover first; then to separate power amps; then from each power amp to a single driver for that power amp. The advantages are: 1.) The crossover is operating under a low voltage load, allowing it to be much more sophisticated (even allowing it to include DSP); 2.) The power amps can be sized, optimized to each driver (some actives use different power amp topographies for woofer/tweeter); 3.) The power amps are given a simpler/more direct "sensing" of the load to greatly improve synergy; 4.) Dynamic performance can be greatly enhanced; 5.) Frequency response is flatter (an epiphany in itself); 6.) Bass can go deeper/louder.
About 16 years ago I auditioned Paradigm Studio 20 v.2 ($800/pair, 2-way stand-mounts) against Paradigm Active 20 ($1600/pair, same drivers/cabinets). It was no comparison. Passersby thought the Active 20's were Paradigm Studio 100's ($2200/pair floor standers, but frankly the imaging was better with the Active 20's). Dynamics weren't comparable. Frequency response was a revelation. And the bass gobsmacked everyone in the room. You can search the web for further discussion. And note that you can buy/build active speakers of your own, but it's big, somewhat complex, and more expensive.
Keep in mind that roughly 98% of all recording/mixing studios use active speakers to have an accurate portrayal of the sound, while frankly the bottom line for home audiophiles is to be entertained with their chosen colorations.
BTW my two pair of speakers in my audio system are single driver (Brinesacoustics.com M18-F200) and active 2-way (Dynaudio BM5 Mk III).
