I am fully aware of the mathematical definition of either and how they relate to conic sections.
Still, I wonder whether there is an easy way to see that these are fundamentally different curves. By "easy way" I mean just by looking at it or by an easy "experiment" or something (no calculation).
Note that I'd be satisfied with any difference. I don't need a proof that they are actually a hyperbola/parabola.
Concretely I have a light shining on a wall (a light cone intersecting with a wall) and would like to interest a 4th grader to the different conic sections that you can see there.
The most easily discernable visible difference that I can think of is that hyperbolas have asymptotes: A pair of (non-parallel) straight lines that they come closer and closer to as you move away to infinity.
For a parabola, on the other hand, the two "ends" come closer and closer to being parallel as you move towards infinity, and there is no straight line they approach.