Why is the Solar Helical (Vortex) model wrong? Could someone give a lay description of why the Solar System's depiction of motion as a vortex is wrong?

ref: Youtube video The helical model - our solar system is a vortex

Please don't take for granted that because of my previous question about why planets or asteroids don't have cometary tails, that I believe the 'tails' shown in the video are real. I do not. I know that the tails as shown are simply a graphic representation of the path 'flown' through space by the planets. It seems that the solar system we inhabit has to be moving through interstellar space in some fashion. If the paths of the planets could be shown in a video, it seems to me (a lay person) that there would have to be motion 'forward' in addition to the eliptical path around Sol.
It isn't correct, because a vortex is not a helix, and so while the planets do trace a helical path as they move through the galaxy, this is not evidence of a vortex. 

Yes, the sun actually is moving through space, as it traces a path around the centre of the galaxy. The whole mass of the solar system moves with it, so the planets are not left behind as the sun moves.

Rhys Taylor and Phil Plait have comprehensive smackdowns debunking this vortex idea and other misunderstandings/delusions by the author.