The relationship between $K_p$ and $K_c$ is given by the equation: $K_p = K_c(RT)^{\Delta n_g}$, where $\Delta n_g$ is the difference between the sum of the number of moles of gaseous products and the sum of the number of moles of gaseous reactants. For $K_p$ to not be equal to $K_c$, $\Delta n_g$ must not be equal to zero. Let us calculate $\Delta n_g$ for each option:

1. $2NO(g) \rightleftharpoons N_2(g) + O_2(g)$; $\Delta n_g = (1 + 1) - 2 = 0$. Hence, $K_p = K_c$.
2. $SO_2(g) + NO_2(g) \rightleftharpoons SO_3(g) + NO(g)$; $\Delta n_g = (1 + 1) - (1 + 1) = 0$. Hence, $K_p = K_c$.
3. $H_2(g) + I_2(g) \rightleftharpoons 2HI(g)$; $\Delta n_g = 2 - (1 + 1) = 0$. Hence, $K_p = K_c$.
4. $2C(s) + O_2(g) \rightleftharpoons 2CO_2(g)$; $\Delta n_g = 2 - 1 = 1$. Since $\Delta n_g \neq 0$, $K_p \neq K_c$. Therefore, the correct option is the reaction $2C(s) + O_2(g) \rightleftharpoons 2CO_2(g)$.