Are orbits equal if and only if stabilizers are conjugate?
The answer is no.
Let the trivial group $G=\{Id\}$ act on $X$ containing at least two different points $x$ and $y$. Then $\operatorname{orb}(x) \neq \operatorname{orb}(y)$ but $\operatorname{Stab}(x)=\operatorname{Stab}(y)$.
More generally, invariant points (i.e. with stabilizer equal to $G$) are not in the same orbit but have the same stabilizer.