Q:

Poisson distribution or Normal distribution

If it is needed to generate randoms in [N, M] range, but with more numbers close to avg (N <= avg <= M), which is better to use:

poisson_distribution or
normal_distribution?

Seeing at examples at cppreference pages (at bottom of the pages), they both generate what is needed:
poisson_distribution at point 4:
0 *
1 *******
2 **************
3 *******************
4 *******************
5 ***************
6 **********
7 *****
8 **
9 *
10 
11 
12 
13

normal_distribution at point 5 with standard deviation 2:
-2 
-1 
 0 
 1 *
 2 ***
 3 ******
 4 ********
 5 **********
 6 ********
 7 *****
 8 ***
 9 *
10 
11 
12

What to choose? May be something else?

A:

Neither choice is great if you need the outcomes on a bounded range.  The normal distribution has infinite tails at both ends, the Poisson distribution has an infinite upper tail.  At a minimum you'd want a truncated form of one of them.  If you're not truncating, note that the normal is always symmetric about its mean while a Poisson can be quite skewed.  The two distributions also differ in the fact that the normal is continuous, the Poisson is discrete, although you can discretize continuous distributions by binning the results.
If you want a discrete set of outcomes on a bounded range, you could try a scaled and shifted binomial distribution.  A binomial with parameters n and p counts how many "successes" you get out of n trials when the trials are independent and all yield success with probability p.  Make n = M - N and shift the outcome by N to get outcomes in the range [N,M].
If you want a continuous range of outcomes, consider a beta distribution.  You can fudge the parameters to get a wide variety of distribution shapes and dial in the mean to where you want it, and scale+shift it to any range you want.

A:

You can center both distributions in a point that suits your needs. 
But if M is small, then the Poisson distribution has a 'fat tail', that is, the probability of getting a number above M is higher compared to the normal distribution. 
In the normal case, you can control this chance via the variance parameter (it can be as small as you want).
The other, rather obvious difference is that Poisson will onli give you positive integers, whreas a Normal Distribution will give any number in the [N,M] range.
Plus, when [N,M] are large enough, the Poisson converges to a Normal distribution. So even if the Poisson is the right model, the normal approximation won't be so inaccurate.
With this in mind, if the numbers do not simulate a counting process, I would go for the Normal.

