The basic shape of a Poisson distribution changes unlike a normal distribution which is always symmetric.
The Variance of the Poisson distribution is equal to the mean while in a normal distribution these are two separate parameters.
A Poisson distribution is discrete while a normal distribution is continuous, and a Poisson random variable is always >= 0.
Arguably, there are underlying similarities between Poisson distribution and Normal distribution. The similarities are;
1. The Gaussian and Poisson process are both Levy Processes.
Levy Processes describes the motion of a point whose successive displacements are random, displacements in pairwise disjoint time intervals are independent, and displacements in different time periods of the same duration have equal probability distributions.
2. Both Poisson and Gaussian distributions can be approximations of the Binomial distribution with large N.
3. The Sum of two normal is normal the same happens to the sum of two Poisson.