They changed the icon on quests which unlock content / features to make them stand out. Sidequests with those blue-ish icons unlock stuff. And yes main scenario quests (msq) are the single most important thing to do in this game. Need to progress the story to reach new places and content. It's recommended you just scoop up all the quests you see everywhere. You don't need to save them for giving exp to other classes since there are plenty of repeatable exp sources like levequests, fates and dungeons.