Stupid, Honest Question: What is there to do in Destiny?
* single player
Because of their online implementation, I'd go so far as to say there's barely any "true singleplayer." You can be alone, but you can also find yourself with 3 others in a firefight around the corner. The game is at its best in cooperative play, either against enemies or other humans.
That said, yeah, there's a story campaign to follow, and the typical "loot game" gameplay loop of improving your character to be able to take on tougher content and improve further, etc.
* competitive multiplayer
yes, this all feels very "legit Bungie" and really worthwhile from what I've played so far. It's 6v6 and I've only seen basic modes so far, but it's fun.
* some variation of horde mode?
I don't know about that, but maybe in the sense that there are strikes and a raid that you need to gear/group for.
What makes Destiny unique?
Honestly, lots. Most of the design elements are actually novel outside of quests/mission structure. I think calling Destiny "generic" is the laziest possible criticism. The only game it truly has much in common with is Borderlands, and it differs wildly in tone, art direction, setting(s) gameplay feel, class balance, itemization, multiplayer implementation and focus. Summoning and controlling your "mount" feels awesome. Bumping into and forming hasty incidental teams with random people in the world is awesome. The various ways you can progress your character, including by playing PvP, while also simply seeking out stronger tests of skill, create an addictive dynamic that sits somewhere between Borderlands, Halo, and Diablo-style games on the "dangling carrot" scale.
If you think about the "raid" and idea of gearing up and assembling a good team for "endgame," I'd go so far as to say there really aren't any other games exactly like this. Now, yes, the essence of gameplay is killing aliens with guns and grenades and special abilities, and that has been the case in a million games. But if that turns you off, there's no hope you'll enjoy the game in the first place.
What makes it next-gen? (though, yes, its on last-gen, too)
Meaningless term. Nothing does, never expect any software to be able to honestly characterize itself as "belonging to the next generation." It has the polish you would and should expect of a current generation (XB1 and PS4) game from a reputable studio.