Hardcore are people that aim for the final destination right from beginning. They want to be at the top, even when they're still at the bottom, and spend most of their free time working toward it. They want to be the best, or, at least for the more realistic of them, rub shoulders with the best. When they see someone better, that has something they don't have, they work toward getting it immediately. They will work toward multiple goals at the same time, spending more free time as their goals expand. When they finally reach the top they then spend their free time maintaining their position at the top; getting all the rarest vanity loot; grinding out achievements; working out new strategies with guild-mates; PvP to test/train personal ability; raising other classes/alternate characters so they can swap in other roles/ability sets to complete specific content; scheduling dates and times that they will meet with "their" group to complete specific content; completing all new content as soon as possible to try and claim world-firsts, etc. At some point when they figure out that they have nothing left to work toward and no new content on the horizon, they'll quit the game when something new comes along. Later, when they're bored of their new stuff, they'll look back and see what expansions have come out. Then they'll re-install/re-sub and work toward completing the new content, and they'll stick around until, again, something new comes along to replace it.
Casuals look at the world immediately in front of them and don't worry about the journey ahead. They'll complete content when it's appropriate or even later if it's initially too hard. They don't like being forced into groups with strangers to complete content, but they'll do so if it's necessary to advance the plot. They don't want to grind, and they'll give up on obtaining items/completing sidequests if it would take too long to complete. When they finally find out what the end-game will be, they just shrug their shoulders and say they'll get there eventually. When they finally get to end-game, they complete what content they can and ignore content that's too difficult. They raise other classes if there are good cross-class skills they can use, but they wouldn't even consider raising an alt. They'll grind under duress if the story requires it, and they see party-play as stressful even before they get into a party. At some point when they figure out that they aren't having fun anymore, they'll quit the game without waiting for something new to replace it. Later, when they're bored and looking for something to do, they'll look at the expansions that have come out, shrug their shoulders, and say it's too much effort to re-install/re-sub for something that's not really fun for them anymore.
Most players are mid-core, who mix and match attributes of both hard-core and casual. Maybe they like levelling alts but don't care about end-game. Maybe they don't like party play but they still try to complete party content while solo. They'll grind for good gear so they can take on greater challenges at the end-game, but they'll quit the game when they look at all the content that remains before them and decide that it isn't worth the effort of keeping up what is basically a second job to complete the parts they care about completing. Later, when they're between games and looking for something to do, they might see that some expansions have come out and evaluate what content has been added. They might even re-install/re-sub if they figure there's enough to keep them going for a few months until they can drop it for something newer/better.
tl;dr - Hardcore want more at end-game. Casual want more along the way to end-game. Mid-core want both to have a lot of content.