Let $X$ be a complex smooth projective variety, and $C\subset X$ a smooth curve. Then $C$ defines a cycle $$\beta=[C]\in H_2(X,\mathbb Z).$$ I have a very vague question about this situation:

Q. If $C$ is rigid in $X$, how far is this condition from $C$ being the unique curve on $X$ in class $\beta$?

I would not say that rigidity implies that $C$ is the only curve in class $[C]$ (what about the converse?), but somehow I cannot quite distinguish the two situations: I have a lack of examples in this sense, and this is what my question is really addressing. Also: does the answer to the above question depend on $X$? What if $X$ is a Calabi-Yau threefold?

Thanks!

PS. Feel free to improve the title...

Edit. By "rigid", I mean $H^0(C,N_{C/X})=0$, where $N_{C/X}$ is the normal bundle. I think this is equivalent to $C$ being an isolated point in the Chow variety of $1$-cycles in $X$ (but please correct me if I am wrong).