Goth evolved from punk. Most of the important early goth (or "gothic") bands - the Banshees, Joy Division, UK Decay- had at one point been punk bands, and something of the spirit of punk passed into the early goth movement.
By 1981, British punk had devolved into two main groups: Oi! and the anarcho movement centred around Crass. Neither side had much to offer musically, and some journalists looked to rising goth bands like Southern Death Cult as keeping the spirit of punk alive. Hence the tag Positive Punk which was applied to some of these bands.
Musically, many of the early goth bands differed from punk in that they'd discarded the use of buzzsaw guitars, instead using tribal drums with shards of guitar thrown over the rythm section. The lyrics and imagery were also changing, tending towards the dark and morbid rather than the straightforward and political. However, there was no clear dividing line, and many of the early goth bands played with and/or shared labels wirth punk bands.
From a fashion point of view, goth still shared much in common with punk, particularly as regards spiky hair, ripped clothing and mohawks (though the classic goth mohawk tended to be black and a lot wider). Since the Banshees were for a while still regarded as a punk band, the "Siouxsie-clone" look was by no means restricted to proto-goth girls, and a lot of the early gig-going crowd looked a lot more punk than goth.
As regards contacts between the subcultures, for a long time goth was part of punk, and it was a while before it became a subculture in itself rather than a subdivision of punk. After about 1983, however, goth had emerged as a separate subculture and became increasingly separate from punk in both music and style.
As a snapshot of the punk scene at the time, here's a couple of quotes from Abbo of UK Decay in US magazine Flipside in April 1981:
Al: How is the audience for you and the bands you play with?
Abbo: Well Crass have a very heavy punk audience, Killing Joke is slightly more sophisticated, a lot of poseurs you might say - can't really tell what kind of an audience us and Killing Joke draw, its a mix between hardcore punks, normal punks, straights, hippies, trendy punks... In London there's still a really good punk scene, there's a lot of places that are really tuned in, other places it's dead.
Abbo: Yeah we used to be very pacifistic, but now only in our views on war and stuff, it's basically impractical to be a pacifist - you've got to get off your ass and do something about it... We can't even play in our hometown because we are associated with punk, smaller bands get away with it if they keep it low key, we try to, but in London there's always a place to play, if one place closes down there's always another opening up.
Al: There are a lot of punk factions...
Abbo: Oh yeah, there's the Spandau Ballet/Stray Cats type of bands, Adam and the Ants, there's us, Theatre of Hate, Wasted Youth (not L.A.'s) bands on a different level, then there's the Cockney Rejects, Sham, Upstarts...
Spon: Football, Oi Oi bands...
Abbo: Totally three factions that don't mix. Over here we've been playing with the hardcore punk bands, which is strange.