You can hear the difference, yes. Does it matter? No, not in the slightest... because the difference is absolutely tiny compared to what you can do by moving your bow a centimetre up and down the string, or by changing the angle you pull pizz at, or any number of other things. And, adjustability buys you so much in terms of dealing with the weather, changing string height to cope with the demands of what you're playing that night, whatever. So adjustable is the right answer.
I think a lot of people start obsessing over tiny variations in gear; the big things matter, like what strings to use, fingerboard profile, string height, soundpost position. Don't worry too much, though, get the instrument serviceable and then get your playing right. Gary Karr playing a beat up ply bass with 20 year old strings will still sound like Gary Karr. Actually, I can say that for sure, because I've heard him do exactly that... borrowed a rental bass off a student, and played like he always does.