I think it's down to a few things; firstly, the mono signal path. Secondly, the lack of multiple effects seen in the music of the day. Thirdly, the tuning, either slightly off by a few Hz or drifting over time.
However, the biggest reasons were nothing to do with the synths, and everything to do with the recording; the warmth of the tape and it's natural saturation; the mixing, which was done from less tracks to begin with, and the slightly looser configuration, especially in the quantise. The slightly more human feel in a lot of that stuff came because ot was played live with a minimum number of retakes, with no comping, just like any Rock record.
Any synth, old or new, analogue or digital, can create the same sounds if the user is technically proficient with their chosen instrument. It's more a feature of the mix and the recording path than the gear, I feel. of course, forum addicts will continue to argue this digitally until the analogue cows come home, and it's a shame to see so many folks buy analogue synth after synth, when it's the overall approach to recording that needs the attention!