The Jordan-Polya numbers are the products of factorial numbers A001013:
1,2,4,6,8,12,16,24,32,36,48,64,72,96,120,128,144,192,216,240,256,288,384,432,480,512,576,720,768,864,960,1024,1152,1296,1440,1536,1728,1920,2048,2304,2592,2880,3072,3456,3840,4096,4320,4608,5040,5184,5760,...
It seems that the difference between two consecutive Jordan-Polya numbers itself is a Jordan-Polya number. Is this true and if so, are there a proof of this somewhere?
This is false. Using the data in your list...
$$ 4320 - 4096 = 224 \text{.} $$
The next few violations of the conjecture are: \begin{align*} 8192 - 7776 &= 416 \text{,} \\ 8640 - 8192 &= 448 \text{,} \\ 16384 - 15552 &= 832 \text{,} \\ 17280 - 16384 &= 896 \text{, and} \\ 25920 - 24576 &= 1344 \text{.} \end{align*} So, while it is common in the first few instances that violations include a power of $2$, this is not true for all violations.
There are $6853$ Jordan-Polya numbers up to $20!$. Computing the gaps and splitting up the first $6800$ gaps into blocks of $100$ gaps ("centuries"). We plot the number of gaps conforming with the conjecture.
After initial high levels of conformance, by the tenth century (thousandth gap), the likelihood of a Jordan-Polya number gap being a Jordan-Polya number seems to be around $50\%$ and is, at least for the range covered, fairly stably so.
Repeating up to $30!$, there are $91\,802$ Jordan-Polya numbers and $53\,065$ gaps are violating. The plot by centuries ...
... suggests that conforming instances become rarer as one proceeds.