Would you believe I've drafted and tossed three separate responses to this thread? I don't think I have a short answer to it, but I'll try and create one.
The two terms aren't separate to me. It's a game in which you play a role; a character who has the power to affect change. If you don't get into the character (the roleplay) than the game (the talking, meeting, killing) doesn't mean anything. The two aren't separate, they're permanently linked.
If I distill everything down to a pile of stats and set you loose in a dungeon, you aren't role playing. You might make the argument that modern parlance for the RPG is directly related to the improvement of a character over time via stats & equipment, but that's a poor definition used by reviewers of videogames to describe a genre. In that environment, you can succeed with a pieced together story and a whole lot of looting & dice chucking.
If I remove all of your stats and simply leave you standing in a persistant world without a means of interacting with it, we can roleplay until we're blue in the face, but you'll never know the full extent of your ability to affect the world around you. You can't, because I haven't defined it (the game).
So a role playing game is the marriage of two concepts; ancient storytelling, where people would 'play out' the roles that the narrator gave, coupled with combat training, as the boy playing the Tiger wrestles with the Hero. We've grown up and taken the idea with us; now we have rules arguments and min-maxing discussions, and yet we still manage to scream "I hit him!" "No you didn't!" across a crowded room.
I know there are people out there who barely roll dice in a session; they're pure role players and if you took those dice away, their game really wouldn't change. And there are people who have never once thought about their character history - or even what their character thinks about - beyond ale & whores. For me, the balance is in giving people what they want, which is usually loot & combat, while doing what I want, which is telling a story in a world I created.