Runebound isn't so much a game as it is a gaming system. With so many different rules tweaks and expansions (even a second edition) it is a hybrid role-playing game/board game.
Just like RPGs there is a lot of customization. Each group can play Runebound differently. Some groups play as multiplayer solitaire. Others add class decks for more interaction. Don't like the default fantasy setting? Play in Arabian deserts with Sands of Al-Kalim. Tired of the same old monsters, use a challenge deck expansion. Play an all against one game with Midnight expansion. Sail around with the Isle of Dread. Don't like fighting Dragon Lords, there are Giant Lords, Storm Lords, members of the dire Cult of the Rune, and so forth available in adventure variants.
Arkham horror is based on the Call of Cthulhu RPG. There are 8 different Ancient Ones in the base game. Everyone always works together to stop the Ancient One from doing whatever evil it wants. The Ancient Ones do give a bit of variety to the game, but each Ancient One is basically trying to open Gates. The differences between them are minor game effects and different number of gates needed to open.
I like Runebound better than the base version of Arkham Horror. There always seems to be something to do to get closer to the win in Runebound. Maybe save up money for that nice item in Dawnsmoor (it is nice to see what you will get so you can try to manage your money). In Arkham, I frequently find myself without a viable option on my turn (maybe I'm just bad at Arkham, but having no good moves is boring).
You can also level up your Runebound character and improve their stats. What makes the game work is leveling your character, moving on to harder encounters (as you level). And trying to keep ahead of the other players leveling. In Arkham there is no leveling. Monsters just come out at random (easy and hard mixed).
How do they compare on story? There is an effort at giving flavour text which has a story to encounter cards in Arkham. These cards are randomly drawn and typically give a boon or just bad stuff depending on the location you are in (and if you need to roll against stats).
In contrast, Runebound has story progression. It does have random stuff (not tied to location) called Encounters, but it also has Events. Events are global effects which give the game a progression, and sense of story. Even games where Event text is ignored have this sense of progression (seeing the Cult of the Rune spreading its corruption across the land, experiencing increasingly worse Storm cards, and so on).
This is not to say that Arkham is a bad game. I love it with expansions, but the base game is sorely lacking fun for me.