Partly it has to do with what you started with. I started on classical and jazz piano for many years, unweighted action always feels wrong to me. Traditional analog synthesizers had a history of light, inexpensive keybeds (most not having velocity anyway), and many of the now-classic mono-synth lead styles of playing are a bit better suited to unweighted keyboards. Rhodes and Wurli's, being amplified acoustic instruments (like an electric guitar), operated in a similar fashion to a piano, though with a bit lighter touch. Organs, on the other hand, are quite a bit different, and though they have a bit more resistance, are similar in design to synthesizer keyboards. People who start on church organs or B3s take up unweighted keys very quickly.
Although, if you want an objective reason as to why weighted keys are better: more pressure means that you can be more nuanced in how dynamic you are playing. Weighted keys give quite a bit more resistance, which means you have to work a little to make them loud, and can adjust the sensitivity pretty easily. Unweighted, in my experience, you have to have INCREDIBLE control to be able to play fine dynamics, because they pretty much always just break away under your fingers and go down at similar speeds, so it's incredibly difficult to be consistent. Also, it's incredibly easy to accidentally hit unwanted notes when playing unweighted, as the keys go down with the slightest touch. But it's no different really than the fact that a cheap violin is harder to play than an expensive one.
So my advice is, if you plan on playing more vintage synth/organ material unweighted keys will do you fine, but for piano/e-piano or modern synth styles that emphasize dynamic change, you're going to want to go for weighted. Yamaha and Casio make some DAMNED GOOD electric pianos for between $400-$500 that I can highly recommend for that direction (modern Casio is a bit better, IMO).
TBH: the only real advantage I can ever see for getting an unweighted is the price and the weight. "Weighted" keyboards aren't just a catch term, they have to actually use weights in them for the action to work correctly, and it makes the board heavier. Though Fatar and Casio have been able to bring that way down to the 15-20lb range, they're still 2-3x heavier than unweighted boards, and some go up to 60-70lbs!