jmc

Subtracting the two equations gives: $$\begin{array}{rl}(2012a+2016b)-(2010a+2014b)& =2020-2018\\ 2a+2b& =2\\ a+b& =1\end{array}$$Multiplying this equation by 2010 and subtracting the resulting equation from $2010a+2014b=2018$ gives $$\begin{array}{rl}4b& =(2010a+2014b)-2010(a+b)\\ \Rightarrow 4b& =2018-2010\\ \Rightarrow 4b& =8\\ \Rightarrow b& =2.\end{array}$$So $a-b=(a+b)-2b=1-4=\boxed{-3}$.