I know a theorem that any union of connected sets is connected if there is a non-empty intersection. I can also think of a counterexample to the converse: take the interior of the unit circle and the circle boundary in $\mathbb{C}$.
It seems to me that if two connected sets are disjoint, whether their union is connected relates to limit points somehow, as points on the circle boundary are limit points of the interior. Is there a simple exact condition of when the union of two (or finitely many) connected sets is connected?
In Kelly's book on general topology you can have a sightly more general result: If $\{A_\alpha:\alpha\in I\}$ is a collection of connected sets in a topological space, and for any $\alpha,\alpha'\in I$ either $\overline{A_\alpha}\cap A_{\alpha'}\neq\emptyset$ or $A_{\alpha}\cap\overline{A_{\alpha'}}\neq\emptyset$, then $\bigcup_{\alpha\in I}A_\alpha$ is connected. Check that book for a proof and other details.