Alt (short for alternative) bars might technically be flat in that most models have no rise to them. However, they have much more back sweep, and more bendy parts. They often also have more than one hand position. Alt bars generally can accept standard mountain bike brake levers and shifters. Examples of Alt bars include Velo Orange Crazy Bars, Soma Eagle Bars, Jones Bars, Surly Moloko, On One Mary, etc.
Flat bars are the ergonomic equivalent of wearing high heels on a hike. Flat bars were born outta motocross, where riders needed a wide, honkin’ lever (the handlebar) to keep the heavy front wheel and fork combo from flopping around in technical, loose terrain. All this while riding at a much higher speed, on a much heavier vehicle than a bike. Early mountain bikes used motorcycle or moped handlebars and brake levers, not because they were the best thing to use, but because they were the only kind available. The alternatives were cruiser bars (these would bend immediately when taken off-road), or road bars, which at the time were way too narrow (and flimsy) for off-road use. The only thing early mountain bikers could really ride off-road with was moto-bars.
People are creatures of habit. A boring cliche, but it’s important to keep in mind when you consider why certain things that shouldn’t stick around, do stick around. Flat bars are still manufactured because the first mountain bikes had them. That’s the same reason why mountain bikes still ran triple cranks up till a few years ago, why bike geometry was so standardized for years, etc. But just like it doesn’t make sense for everyone to wear shoes with buckles and powdered grey wigs just cause ole’ G-Double-Ya did, it also doesn’t make sense for most contemporary mountain bikes to keep moto style bars as the primary handlebar option. Why? Because just like powdered wigs, moto style flat bars are not comfortable.