Marshall Thomas said:
How do capital provinces differ from non-capitals? How do core provinces differ from non-cores?
Capital provinces have a -2 modifier to Revolt Risk, and they can never be taken in a peace treaty unless they're the only province remaining (force-annexation). Also, it is the source from which overseas tax penalties and frontier bonuses are derived. Finally, the continent the capital is located in can affect certain events, as well as growth rates in certain overseas colonies; for example, if a country's capital is in Europe, many of its colonies in Africa will have a massive growth malus, whereas a country with its capital in Africa wouldn't suffer from the same malus (in Africa or even in Europe).
Core provinces are considered a legitimate part of your country (or close enough for gameplay/historical purposes). If another country owns one of your core provinces, you will have a permanent Casus Belli on them, and you will never suffer a BadBoy hit by taking them in a war (unless it involves a force-annexation). They also suffer less War Exhaustion.