I am very ignorant about all but the most basic concepts of chemistry, but I'm very curious, so I'm going to ask this question anyway. What is the significance of creating synthetic elements that don't last for any length of time?
The simple short answer is "to prove it can be done."
But more excitingly, the answer is that one can never tell what useful discoveries might result.
Until the early 1940s the periodic table of elements ended with uranium.
Experiments in the 1940s to 1960s led to the manufacture of laboratory quantities of neptunium and plutonium, and eventually americium (95), curium (96), berkelium (97) and californium (98), whose physical properties could be determined. Beyond that, it is true that higher actinide elements have proved to be unstable with very short half lives for most of their isotopes.
Yeah, same reason people climb mountains. Because they're there.