Ability Scores in D&D Next have a number of problems, most of which are carried over from previous editions.
1) Unintuitive: Just as THAC0 (To Hit Armor Class Zero) made mathematical sense but was difficult for new players to remember, the ability scores are pointlessly confusing. It makes no sense that an 18 Strength would give you a +4 bonus, or that a 3 Strength would give you a -4 penalty. Ability scores should provide a bonus equal to the score.
2) Inequality: Strength, Dexterity, and Constitution apply to a large number of important things, whereas Intelligence, Charisma, and (to a lesser degree) Wisdom are only important if it's your class's primary attribute, which makes that attributes weaker. Previous editions have tried to balance out this inequity by making Strength bonuses more difficult to get (I'm looking at you, 3.5 Half-Orc) and Str/Dex based classes weaker. This actually ends up messing with balance in innumerable ways, and punishes players for making roleplaying decisions (I want my Rogue to be a dashing leader. Wait, Charisma sucks. Never mind, I'll just use Dex/Con like I'm "supposed" to). Balance between attributes can be accomplished in a wide variety of different ways - Action Points, additional Reactions, bonuses to key checks (Initiative, Aid Another, Attack of Opportunity, whatever).
3) Lack of Variance: Variance is a measure of how far a set of numbers is spread out. To the degree that variance is determined randomly (by rolling a d20) is the degree to which a game is luck based. For example, an 18 Strength Human should win an arm wrestling almost all of the time against a 3 Strength Halfling. But in D&D Next, the Halfling will win 16.5% of the time. So it makes more sense to have a broader range of ability scores. For example, if 18 Strength gave you an +18 bonus to Strength checks, then the 3 Strength Halfling would only win 2.5% of the time. It also means that you can set a broader range of Difficulty Chances (DC's) for various tasks, with high scores having a much easier time with simpler tasks without the need to invest in a Skill/Trait/Feat (+5ish?) to improve it, and low scores being unable to complete more difficult tasks unless they do so.
I would also say that actual gameplay choices (which abilities/spells/powers to use, movement, positioning, etc) should carry high bonuses or penalties which stack, because doing so makes those choices meaningful. You can do so in a number of different ways without resorting to the jumble of charts from previous editions, or the ultra simplification of the current Advantage system. For example, each Advantage could provide you with a d6 bonus to your roll (or a flat +3 bonus, or something else), and each Disadvantage could provide you with a d6 penalty. Thus, if you had 3 Advantages and 1 Disadvantage, you would roll 1d20 + Ability Score + Trait/Feat + 2d6. This gives you fairly equal weight to luck, ability, training, and choice. And it's quite easy to adjudicate, without resorting to any secondary charts.