Great question! Its super simple, it is ONLY the length of the hydrocarbon chain, a better quality natural rubber has really long chains (10,000+ atoms long) that last a LONG time and are VERY stretchy. Synthetic rubber (or plastic) is shorter (1,000 atoms long) and doesn't last as long.
Thats it, it is the exact same "product" just a chain that gets longer and longer and changes its physical properties as it grows.
It's more than just the chain length.
The most common synthetic rubbers are styrene-butadiene copolymers. Natural rubber is polyisoprene. While it is true that shorter chain synthetic polyisoprene is available, it is a much smaller part of the market than styrene-butadiene.
Genuinely interested here. If natural rubber and plastic are exactly the same thing, then why is natural rubber being advocated as a much more environmentally sound alternative? Would'nt the dust from natural rubber tyres be just as problematic as the plastic is now?