Handwound and machine wound are really bad terms to describe what people think they mean. Typically a "handwound" pickup is still wound by a machine, in the sense that the bobbin is being spun by a motor, but a human is guiding the wire back and forth manually instead of letting a machine do it for them. Hence the term handwound.
Handwound pickups are sometimes called scatterwound, which is probably a more accurate term. The assumption is that a human will be less consistent at guiding the wire than a machine, and the wire will zigzag back and forth more aggressively or more randomly on a handwound pickup versus a machine wound one. That's a bit of a leap, since there are a lot of humans that have been winding pickups for a long time who can easily create a very evenly wound pickup. And, it's trivial to program a machine to "scatterwind" a pickup exactly like a human might do. So although the handwound vs machine wound terms are convenient, they're not really accurate.
So that leaves us with a different question. Is there a difference between a scatterwound pickup (aka handwound) versus one where the windings are laid down nice and evenly and tight (aka machine wound)? Yes! There definitely is. There are clear physical differences, at least. A pickup coil with a lot of scatter and randomness will be less dense. The wire criss-crossing itself will result in less wire packed into a given physical space. If the pickups are being wound to a certain turn count, the overall size of the coils will be different. if the pickups are being wound to a specific resistance, the turn count will be different. These physical differences do translate into slight differences in the signal that's generated. The coil being less or more dense means the wire is physically in a slightly different portion of the magnetic field, so the signal will be stronger or weaker overall. And the density can cause other (probably less significant) differences since the electrical properties will be slightly difference (in terms of impedance and capacitance within the coil).
Does that translate into a difference in tone? Arguably, yes - although there are enough variables involved that it gets cloudy very quickly in terms of describing the difference, and it would be misleading to try to make sweeping statements. If you're comparing a handwound pickup vs a machine wound one, do you know what all the differences between the two pickups are? Probably not! So, attributing a difference in tone between the two is difficult, at best.
As you noted though, handwound pickups are typically treated as more desirable. In my opinion, this has more to do with the culture and tradition of pickup making than anything else. Most people attribute handwound pickups with the early days of electric basses, and hence it falls under the trope of "well, if this is the way it was done back in the day, it MUST BE THE BEST WAY!!!" There are a lot of people who have a lot of money and emotion invested in Musical Instrument Tradition, and those people will fight tooth and nail to persist the notion that it's somehow impossible for anything bass-related to ever get better than the way things were done in the past. It's okay if you like traditional pickup styles of course, but I think it's a stretch to automatically assume they're better.
So, with respect to your question - yes, there will be differences, but the differences don't inherently make one option better than the other, since your personal preferences will play a role in determining which one you like better - and both pickups will be "good" otherwise.