For 2 independent binomial distributions, X~ B(n, p1) and Y~ B(m, p2)
Generally, X+Y does NOT follow a binomial distribution. The special case when X+Y follows a binomial distribution is when p1=p2 = p. Then X+Y~ B(n+m,p)
X-Y does NOT follow a binomial distribution.
If X and Y fulfill the conditions for using Normal approximation to Binomial, then X and Y can be approximated individually using Normal approximation.
For example X~ B(70,0.6) and Y~ B(65,0.7).
X~N(42,16.8) approximately and Y~N(45.5,13.65) approximately
Then X+Y~N(87.5,30.45) approximately. Continuity correction applies.
and X-Y~N(-3.5,30.45) approximately. Continuity correction applies.