"Instinct/intuition" are, for example, when you just know someone is watching you, even though you can't see them. It could be that unsettling feeling you get sometimes when your partner is late home from work and, instead of just thinking that they are late because of bad traffic, you're convinced they have been in a car accident (of course, most of the time, it turns out that you were wrong, but you still had that feeling). Baby animals feed from their mothers by instinct - they have not been taught that they should find a teat and suckle on it, yet they know to do this almost the instant they are born.
"Common sense" is knowing not to stick your fingers in a plug socket because you will be electrocuted. If you had never seen a plug socket before, you wouldn't know by intuition that it was dangerous to stick your fingers in it. Those of us who know that it's dangerous, know because we were probably told when we were very young. It's common sense to look both left and right before crossing the road as a pedestrian - it's just something that it is sensible to do.