When you think of what musical timbre distinguishes a song from the 1980s, the synthesizer springs first to mind. If a pop song these days wants to "give it that '80s vibe," they are undoubtedly going to throw in a synthesizer part.
While the synth did play a central role in the orchestration of the period, it was usually paired with another instrument that sounded more traditional and organic, or at least a vocal delivery that sounded soulful rather than robotic.
This contrast of timbres relieved the song from having only a futuristic, artificial sound texture. The synth suggested the eerie, uncertain future of the modern world and its strange new technologies, while the traditional instrument (or soulful voice) provided a root in more familiar and comforting sounds.
The traditional instrument did not have to belong to the culture that the band came from — a foreign one could equally convey the impression of traditional and organic, as well as creating an exotic atmosphere. Especially within New Wave, the bands tried to strike a balance between the futuristic and the primitivistic.
There are too many examples to cover in a single post, so I'll focus things by only looking at the pairing of synthesizers with the marimba as the traditional, organic instrument.
It has its own melodic line in "Loving the Alien" by David Bowie, and the mixture comes from layering multiple instrumental parts.
A more effective use of contrasting timbres can be heard in the main riff for "Love My Way" by Psychedelic Furs, where the marimba's line hangs uncompleted until the synth chimes in for two slow eerie notes. I'm not sure why this succeeds better than the parallel layering approach — perhaps the shift at the end of a sequence strikes us more as a transformation of the organic into the synthetic.
Similar to one instrument completing the phrase of the other, a call-and-response pattern is used in "Sister of Mercy" by Thompson Twins. Both the marimba and flute-y synth have their own little riffs, and they alternate back-to-back. This is almost as memorable as the Psychedelic Furs approach, but they don't sound as intertwined because each part plays a self-contained phrase, whereas the two parts in "Love My Way" don't stand on their own and only work in combination.
Today when you hear the synth, it's more a descendant from electronic or techno music from the '90s and after, where the sole focus is on the futuristic, artificial, and hi-tech. It lacks any contrast with an organic instrument, and the vocal delivery is sure to be mechanical, robotic, and auto-tuned as well. Unlike the pleasingly surprising mixture of futuristic and primitive from the '80s, these techno-derived synth parts are more like boring gadget porn for music addicts.
We can note a similarity between New Wave songs and Art Deco architecture of the Jazz Age, which also aimed to wed primitive textures and exotic iconography with modernistic design plans and materials. Techno music's analog is more like the World of Tomorrow and Space Age styles of the Midcentury, which were entirely forward-looking and meant to dazzle the audience with hi-tech novelties.
Both New Wave and Art Deco arose from outgoing, rising-crime climates, while techno and the World of Tomorrow arose from cocooning, falling-crime climates. Based on other cases, my hunch is that people in an outgoing social mood prefer high contrast in their culture, as opposed to cocooning people for whom that would be too stimulating, and who therefore prefer lower contrast.