For 2 independent Poisson distributions, X~Po(ʎ) and Y~Po(µ)
X-Y does NOT follow Poisson distribution.
If X and Y fulfill the conditions for using Normal approximation to Poisson, then X and Y can be approximated individually using Normal approximation.
For example X~ Po(30) and Y~ Po(20)
X~N(30,30) approximately and Y ~N(20,20) approximately
Then X-Y~N(10,50) approximately. Continuity correction applies.