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Active speakers can make the most of amplifiers that are almost directly connected to the speaker and matched to the resistance of the driver+wiring. In that way a well designed active speaker can use amplifier power more effectively. If it's well designed it should outperform the speaker+amp combination.
Passive + Amp means that the speakers still work if the amp stops working, you can tailor the amp power and flavour to suit your budget etc. The amp can get very hot and use massive heat sinks because it's not constrained by the speaker box.
Those are some basic differences.
Airhead
All of those things. Budget isn't massive, nor is the room. Ultimately I'm not an audio connoisseur, but if one option is better than the other I'd opt for that.
Volumio on raspi has cost me the same £30 as a Chromecast, but can also play from a hard drive or internet radio, and integrates natively into my smart home hub interface.
With raspi i can get a DAC hat and run it directly to active speakers, or a DAC+amp hat and run it to passive speakers, I'm just curious to hear why one would opt for active vs passive speakers?