First note that $10xy-x^{2}-y^{2}=2(x+y)^{2}-3(x-y)^{2}$ and therefore if a positive integer can be represented by this form it can be represented by the other form. Also note that $6=2.3^{2}-3.2^{2}$ and $10xy-x^{2}-y^{2}=8xy-(x-y)^{2}\equiv 0,1\pmod{4}$ so 6 cannot be represented and thus the number of positive integers of the form $2x^{2}-3y^{2}$ is bigger.