Difference between means

The difference between any two independent quantities X1 and X2 has a distribution with

Applying this to the difference between the means of two random samples,

If the distributions are normal in each group, ...
... the sample means are normal, so their difference also has a normal distribution.
Otherwise, ...
... the two sample means are approximately normal if the sample sizes are large, so their difference is also close to normal.

Irrespective of the distributions within the two groups,