- The origin story is different. Although the justification for becoming the Punisher is the same (loss of loved ones); the circumstances are completely different. In both cases, the circumstances heavily related to the main plot.
- Completely different main plot. The movie focuses on a local crime syndicate in a city. The TV show focuses on military/CIA/conspiracy theories.
- The movie's main focus is on defeating the bad guys; and the Punisher's questionable methods are hardly ever questioned (by the plot). Almost always, the Punisher responds to his environment, rather than provoking escalation. The Punisher is also very single-minded, and hardly ever qhowsshows signs of a different opinion, or questioning himself.
- The TV show, on the other hand, implicitly questions the validity of the Punisher's methods. While The Punisher himself is of course still focusing on the bad guys, he also drifts between different approaches. The show spends considerable screen time on side characters, whose plot points sometimes don't even relate to the main plot, but they usually indirectly question whether the Punisher is doing the right thing or not (thus making them on-topic for the viewer).
- The movie's characters were very one-dimensional (especially the innocent bystanders). The TV show has created a more gritty yet nuanced reality. Note: I loved the movie to death. I'm not saying the movie is bad, not at all, but the characters were a bit simple compared to now having seen the TV show.
Again, I can draw an analogy to Batman. The 90's Batman movies painted Batman as unquestionably good, and the villains as unquestionably evil. The same is true of the Punisher movie.
But the Nolan Batman movies focus more time on wondering if Bruce Wayne is doing the right thing, and not just using Batman as a way to vent his anger. Similarly, the Punisher TV show questions Frank Castle's state of mind and methods, instead of assuming they are unquestionably good.
About the TV shows
Although the movies are unconnected to the TV shows, and the movies are unrelated to each other; the TV shows are connected.
The Punisher is first introduced in season 2 of the Daredevil TV show, and I highly suggest you watch this before you start with The Punisher. WIthout spoiling things, most of the existing Punisher lore is handled during Daredevil, and The Punisher picks up the story after the events that transpired in Daredevil.
It's not impossible to watch The Punisher (show) before Daredevil (show), but I highly advise against it. The Punisher is somewhat lighter on character exposition of Frank Castle exactly because his personality (and The Punisher's main goals) was exposed during Daredevil.