Let us consider for simplicity the complex semi-simple case. As you mention, every irreducible representation of a semi-simple finite dimensional Lie algebra $g$ integrates to a representation of the corresponding simply connected Lie group $G$. But the representation of $G$ may well have a kernel, a finite central subgroup $H$ of $G$. So we obtain representations of $G/H'$ where $H'$ is a finite central subgroup contained in $H$, but not of other groups with Lie algebra $g$.

So one could ask: given a Lie group $G'$ with Lie algebra $g$, when does a representation $\rho$ of $g$ with highest weight $\Lambda$ integrates to a representation of $G'$? The answer: if and only if $\Lambda$ belongs to the character lattice of a maximal torus $T$ of $G$. (Recall that the character lattice of $T$ is the discrete additive subgroup of the dual of the Lie algebra of $T$ spanned by the differentials of the homomorphisms $T\to\mathbf{C}^{\ast}$.)