We approach this with complementary counting, so we are looking for the number of non distinct values. Suppose integers $i,j\in\{1,2,3,...,100\}$ such that $\frac{i^2-2}{i^2-i+2}=\frac{j^2-2}{j^2-j+2}$ but $i\neq j$. After cross multiplying and a ton of simplification, you get $(i-4)(j-4)=14$. So $(i,j)=(5,18),(6,11)$ since order doesn't matter. Therefore, our answer should be $100-2=\boxed{98}$.