Jurassic Park cares about Dinosaurs, Chaos theory and Genetics. The fantastic failure of every piece of media attempting to recapture the charm of the original Jurassic Park movie is due to failing to appreciate these aspects.
The original Jurassic park portrays dinosaurs primarily as animals. It portrays them sometimes as supernaturally clever and mysterious but always as animals, and every scene with them treats them with the respect that animals deserve. We see an astonishing variety of dinosaur behaviours: hunting, birthing, eating, flocking, dinosaurs inflicted by illness, to name a few examples. Each of these depictions is focused towards portraying them as animals and highlighting how foolish it is to expect them to be anything else, especially to expect them to be theme park attractions.
The dinosaurs don't appear on cue just because the audience is waiting for them, they don't stop breeding because it would be convenient for the park hosts. We see in the guests tour around the park that dinosaurs can make for boring attractions, the Tyranosaurus barely shows itself even when lured out by live prey, and without proper care -such as proper food - animals like the Tricerotops is stuck ill. Not only are they boring attractions, but extremely unruly ones as well: being so poorly understood that trying to contain them ends up impossible.
It's the failure to appreciate dinosaurs as animals that is the eventual reason that the park fails; the other points of failure could have been resolved, there were a million contigency plans, but it's this fundamental lie - that the dinosaurs could be treated as only attractions - that dooms the project.
After the introductory act, there are many action sequences involving dinosaurs where the lives of the humans are at risk. The thrill of these scenes has the purpose of emphasising these as dangerous animals, a danger which has gone unaknowldged during their creation. They are willing to hunt prey and attack what they perceives as a threat, just like any animal. But their existence isn't defined by violence: The T.Rex is perfectly willing to leave the humans alone once it's secured a sufficient meal, and most of the dinosaurs aren't much bothered by the humans being more curious than anything.
The sequels do a great dis-sevice to this portrayal, focusing on dinosaurs as either being spectacles of violence or elegance. In the end, any effort they make to say that these dinosaurs are real living creatures that don't exist solely park tourist's entertainment, is dashed by the fact that they treat them solely as set piece entertainment for the actual audience. We see violent battles between large carnivores which would never take place in real life, we see dinosaurs defined by their brutality or size. This is not the respect you show to an actual animal, which is why the sequels fail to capture this fundamental essence of Jurassic park.
Portraying dinosaurs as believable to an audience was therefore an extremely high priority for the film makers. Although puppets could be used for many sections - and they spared no expense here, by hiring industry veterans - they alone would not be enough, and so CGI was revolutionised, practically invented, to give nearly photorealistic depictions. The actual use of cutting edge technology in the production of the movie complimented the themes of experimental science extremely well, which lead to a kind of mythology around the movie that cannot be replicated.
Jurassic Park was released at the optimistic beginnings of our efforts towards genetic engineering, when it felt like we were only years away from solving the problem of genetic diseases. In case you haven't noticed, much of this dream was mis-guided. The progress of genetic engineering has been much slower than we initally thought, with gene therapy only recently being introduced to the public and the idea of estimating life spans by genetic sequencing being thrown out. Like any scientific and technological revolution, the inital reach of the field ends up falling well ahead of its eventual grasp.
To say the least, we cannot recreate dinosaurs, or any animal, from its genetics alone. There are a billion details, such as the hormonal make-up of the mothers womb, epigentic triggers or learned behaviours, which prevent DNA alone from helping us resurrect an animal.
This isn't to say that genetic theory is useless, but that it's been seasoned with a great deal of nuance in the time since the original Jurassic Park. Sequels have largely ignored these scientific developments - just as they've ignored devlopments in paleontology - as if the orginal movie was made famous by focusing on outdated science, instead of its enthusiatic willingness to guess what the future holds.
Chaos theory is one of the worst communicated ideas in all of science fiction. I sadly have to blame Jurassic Park for much of this, although I'm pinning this blame on Michael Criton rather than Speilburg. Especially in the book, the character of Malcom doesn't go much deeper than gesturing to the entire park and saying: "y'know chaos theory makes this impossible", and when someone patiently asks him how he came to that conclusion he rambles on about nonsenical technical jargon without actually saying anything.
The portrayal of chaos theory throughout the story doesn't go much farther than outlining the butterfly effect: using it to suggest that the ability for nearly imperceptible details to spiral into masssive changes makes the dream of controlling so many unknown variables impossible. No matter how well Hammond sets up feeding routines, emergency protocols or containment procedures, the human errors made in its design are exploited constantly by the forces of chaos generally, and the unknown agency of the dinosaurs specifically. The project is simply doomed to failure.
This broad view of Chaos Theory as a guarantee that something will eventually go wrong is, probably, an oversimplification of a vase field of research. It does work in Jurassic Park however, by synthesising this unknown modern idea - extremely modern by the time the story was written - with the unknown capabilities of the ancient dinosaurs. It's a frightfully modern omen of danger, a warning that the endless drive of technological progress is taking away our ability to choose where it takes us, which makes the character of Malcom a prophet of destruction by the claws of a very old reptilian force.
This view of chaos theory is tied very closely with the concepts of cascading failures and the many points of failure large system possess. The three mile island incident was a massive cause for this line of thinking, and many of the resulting theories about what went wrong would go on to inspire stories like Jurassic Park. Looking at the movie with Three Mile Island is eye opening, especially when you compare it to similar stories written before the incident. Our Lady Of The Sauropods is strikingly similar to Jurassic Park, but takes on the form of a psychological drama instead of cascading failure motif in Jurassic Park.
The sequel stories aren't much able to capitalise on this key theme of the orignal story. The Lost World was written with population dynamics in mind rather than chaos theory - it's an original twist and continues to use mathematics as a prophetic tool, but fundamentally the sequel doesn't use this idea to say much more than "Populations sure grow fast huh?" probably because Crichton knows even less about population dynamics than he does chaos theory. The other attempts to revisit chaos theory use it more as an excuse for why things keep going wrong than to express any original concept, and since the audience is already familiar with the concept it just comes off as lazy writing trying to reference what used to be interesting.
There are really two ways to recapture this element of Jurassic Park:
Although Jurassic Park talks a lot about Chaos Theory, it completely misses two extremely related subjects which also emerged out of the 70's and the beginning of the Cybernetics movement: Ecology and Control Theory.