Not a huge difference. It is more about time. It's about to rain = it will rain soon/now. I'm about to sleep (fall asleep) = I will sleep soon/any minute/now. It will rain = some time in the future it will rain...maybe tomorrow. So, when I look at the clouds and think "it will rain any minute now!" I would say "It is about to rain. Let's go inside"....or in the South we would say "it is fixing to rain! Let's go inside".