^ A little something I drew a few years ago.
Goth is kind of an offshoot of EARLY punk rock and became it's own thing in the 80's with music commonly called "post-punk" which was like a less poppy/dumb version of New Wave. The biggest difference I've come to notice throughout my years of being into alternative-culture is that punk rock's frustration stems from the outside world, a goth is more self loathing and introspective. Also, punk rockers are really closer to hip hop if anything. They hang out in bad neighborhoods, drink 40 oz malt liquor, carry knives and hopefully don't run in "crews" or gangs. Goth people dance to industrial music or "darkwave" these days and are generally more nerdy. You're not likely to walk in on a group of punk rockers playing Dungeons and Dragons.
This is just the bare essentials of each. There's a 1,000 offshoots and sects of each.