Let $X$ be a smooth variety. Then we know that the picard group of $X$ is the quotient $\text{Div}(X)/\text{Princ}(X)$ of the group of divisors of $X$ by the subgroup of principal divisors of $X$.
Now suppose in the case where $X$ is a complex manifold, one has the exponential sequence: $$ 0\mapsto 2\pi i \mathbb{Z} \mapsto \mathcal{O}_X \mapsto \mathcal{O}_X^\ast \mapsto 0$$
Now we know from usual cohomology that one has the induced morphiam $$H^1(X, \mathcal{O}_X^\ast) \mapsto H^2(X, \mathbb{Z})$$ I've also read that in this case $H^1(X, ,\mathcal{O}_X^\ast)$ is the Picard Group of $X$ as well.
My question is how are the two definitions of the Picard Group I've stated above equivalent? Moreover, how does is image of morphism identified as the Neron-Severi group of $X$? The usual definition I have for the Neron-Severi group is the group of Cartier divisors modulo numerical equivalence. Would be thankful if anyone can provide some answers.