Any Lie group homomorphism $\phi : G \rightarrow H$ is determined by the induced Lie algebra homomorphism $d\phi : \mathfrak{g} \rightarrow \mathfrak{h}.$
The derivative $d\phi$ determines $\phi$ on a neighbourhood of the identity of $G$. The image of $\phi$ on the subset of $G$ covered by one-parameter subgroups can be given explicitly in terms of $d\phi$, using the formula $\phi (\exp(tX)) = \exp (t\, d\phi (X))$ for $X \in \mathfrak{g}$. If $G$ is connected then any element of $G$ can be written as a product of elements in one-parameter subgroups, so in this case $d\phi$ determines $\phi$.
However if $G$ is not connected then $\phi$ is not determined by $d \phi$. For example, if $G$ is a discrete group then $\mathfrak{g} = 0$, so $d \phi$ cannot possibly give any useful information.