7. Let $A$ and $B$ be two different subsets of the set $\{a, b, c, d, e\}$, such that set $A$ is not a subset of set $B$, and $B$ is not a subset of set $A$. Then the number of different ordered pairs $(A, B)$ is
7.570.

Notice that, the set $\{a, b, c, d, e\}$ has $2^{5}$ subsets, and the number of different ordered pairs $(A, B)$ is $2^{5}\left(2^{5}-1\right)$.

If $A \subset B$, and suppose the set $B$ contains $k(1 \leqslant k \leqslant 5)$ elements, then the number of ordered pairs $(A, B)$ satisfying $A \subset B$ is
$$
\sum_{k=1}^{5}\left(2^{k}-1\right) \mathrm{C}_{5}^{k}=\sum_{k=0}^{5} 2^{k} \mathrm{C}_{5}^{k}-\sum_{k=0}^{5} \mathrm{C}_{5}^{k}=3^{5}-2^{5} .
$$

Similarly, the number of ordered pairs $(A, B)$ satisfying $B \subset A$ is $3^{5}-2^{5}$.
Therefore, the number of ordered pairs $(A, B)$ satisfying the condition is
$$
2^{5}\left(2^{5}-1\right)-2\left(3^{5}-2^{5}\right)=570 .
$$