jmc

We can see that the maximum positive difference is $16-1=15.$ Some quick calculations show that we can get all values between $1$ and $15.$ $$\begin{array}{rl}16-1& =15\\ 16-2& =14\\ 16-3& =13\\ & ⋮\\ 16-14& =2\\ 16-15& =1\end{array}$$ Therefore there are $\boxed{15}$ different positive integers that can be represented as a difference of two distinct members of the set $\{1,2,3,\dots ,14,15,16\}.$