When are two simulations not a bisimulation?
Because a CCS process is worth a thousand pixels – and it is easy to see the underlying LTS – here are two processes that simulates each other but are not bisimilar:

$$P = ab + a$$
$$Q = ab$$

$\mathcal{R_1}=\{(ab+a, ab), (b, b), (0,b), (0, 0)\}$ is a simulation.

$\mathcal{R_2}=\{(ab, ab+a), (b, b), (0,0)\}$ is a simulation.

$P\ \mathcal R_1\ Q$ and $Q\ \mathcal R_2\ P$ but $P$ and $Q$ are not bisimilar. Why not? Because $P\stackrel{a}→0$ and the only $Q&#39;$ such that $Q\stackrel{a}→Q&#39;$ is $b$ ... and $0$ is not bisimilar to $b$.

Why can they simulate each other, then? Because $P$ simulates $Q$ since it can do everything $Q$ does. And $Q$ simulates $P$ because even if $P$ can go in one $a$-step to a program that does nothing, $Q$ can still do that $a$-step, and that's all it takes to simulate something. The key difference with the bisimulation is really that, as Charles said, you have to relate the same processes with both simulations. (i.e. $\mathcal R$ such that both $\mathcal R$ and $\mathcal R^{-1}$ are simulations)