I don't eat white pudding,now that's a proper Irish thing! In fact I can't even by it where I live!
Potato Farl or boxty or some cold cooked potato leftovers from yesterdays dinner (traditionalists only) is an Irish thing too.
I never have any potato with my English breakfast.
The Irish will also have soda bread where the English breakfast has bread made with yeast as the raising agent.
I generally don't have sausages unless they are of a very high standard (you never know what's in them)!
Both English and Irish breakfast traditionally do.
Baked beans are also often served with an English breakfast as is fried bread.