Core gamers are people who spend lots of hours in a day (more than 2 hours non-stop) playing games. Gaming is often solitary, withe the exception of online friends.
Casual gamers are people who spend a small portion of their day gaming (less than 1 hour non-stop). Gaming is something done between activities or as an activity with friends and family in the same room..
Neither group is more or less relevant to gaming. Both have influence and are an important market to game and device manufacturers.