To get the Mayer-Vietoris sequence from the Cech-to-derived functor spectral sequence one should go one step backwards and consider the $E_1$ sheet rather than $E_2$. Then, applying e.g. theorem 4.6.1 from Godement, Th\'eorie des faisceaux, one gets the long exact sequence

$$\cdots\to E_1^{1,i-1}\to H^i(X,F)\to E_1^{0,i}\to E^{1,i}_1\to\cdots$$

where the last arrow is the $d_1$ differential, $E_1^{1,j}=H^j(U'\cap U'',F)$ and $E_1^{0,j}=H^j(U',F)\oplus H^j(U'',F)$.