It is a standard fact that if two simply connected Lie groups have isomorphic Lie algebras, then the groups are isomorphic themselves. There is an immediate naive (more general) question which I cannot find anywhere answered, so I suppose the answer is negative with simple counterexample, but anyway:
Suppose that $G$ and $H$ are connected Lie groups with isomorphic Lie algebras and fundamental groups, i.e. $\mathfrak{g}\simeq\mathfrak{h}$ and $\pi_1(G)\simeq\pi_1(H)$. Are $G$ and $H$ isomorphic?
Not necessarily.
Let $U$ be the universal cover of $SL(2,\mathbb R)$; it has a normal subgroup $N$ isomorphic to $(\mathbb Z,+)$ such that $U/N\simeq SL(2,\mathbb R)$.
So, the quotient of $U\times(\mathbb R,+)$ by the discrete subgroups $N\times\{0\}$ and $\{e\}\times\mathbb Z$ have isomorphic Lie algebras (both of them are isomorphic to $\mathfrak{gl}(2,\mathbb R)$). But$$(U\times\mathbb R)/(N\times\{0\})\simeq SL(2,\mathbb R)\times\mathbb R$$whereas$$(U\times\mathbb R)/(\{e\}\times\mathbb Z)\simeq U\times S^1,$$which are not isomorphic. However, the fundamental group of both of them is isomorphic to $\mathbb Z$.