Both from exponential family. Waiting times of a Poisson process are exponentially distributed.

Dave gave a good response. I agree

The exponential distribution is that of a real random variable on the positive real line (pdf(t) = c exp*-ct),

whereas the Poisson distribution is defined over integers (pmf(n) = x^n exp(-x*n)/n!).

There is no a priory relationship between the two (because one is over reals and the other over integers), but both characterize different aspects of a Poisson process:

for a Poisson process the inter-point intervals have an exponential distribution; and for *any* time interval T the number of points N(T) in that interval have a Poisson distribution. The latter in fact is a defining property of the Poisson process.

It should be noted that a Poisson process is a stochastic process, so that the sample space is more complicated than that of a Poisson distribution (which is just defined over integers) ..