If number of occurrences per time unit of some event is a random variable $X\sim \mathrm{Poisson}(\lambda)$, then the time between occurrences is a random variable $Y$ which has exponential distribution.
What isn't stated clearly anywhere is that is the parameter of the exponential distribution the same $\lambda$?
Yes, more precisely, if the number of arrivals in an interval $t$ is Poisson $Po(\lambda)$, then the time between two consecutive arrivals is Exponential with mean $1/\lambda$.
Proof:
$$F_T(t)=P(T\leq t)=1-P(T>t)$$
Where $P(T>t)$ means: "calculate the probability that, in a size $t$ interval there are no arrivals".
In other words
$$P(T>t)=e^{-\lambda t}$$
that is
$$1-P(T>t)=1-e^{-\lambda t}$$
which is exactly the CDF of an exponential $\text{Exp}(\lambda)$