Full disclosure: when Wizards announced the D&D Next playtest, I'd had enough. Why buy a new series of expensive books, when I could just design my own?
"Saving throws are obviously important to the game—but was the manner in which they had been previously implemented adding too much detail and complexity to the game?"
Yes. I eliminated saving throws by turning them into skills with alternative uses, and since each skill is based on an ability score, a saving throw could boil down to a simple ability check.
"In past editions, we've used tables big and small to capture all the +1 or –2 modifiers that can creep into the game. Advantage (along with its sinister twin, disadvantage) is easy to remember, simple to apply before or after a roll, and comprehensive enough to devour huge swaths of fiddly modifiers."
I've rolled all the little modifiers into the Difficulty chart. Say you're riding a horse, down a steep slope, through blinding rain, and through a cursed area. What's your attack modifier? The GM decides that, for the average person, this attack is Challenging. Challenging corresponds to a -4 penalty, so that's what the PC adds to his attack roll.
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howandwhy99 : there's a fine line between elegance and simplicity. My take was that an "elegant" game is one that is free of problems, even if those problems are things that work, but players don't like. I definitely encountered this designing my own game: it's so simple that it lacks personality. In my case, though, that's the intent; I want to be able to add rules to make it a different game as necessary.