By flat, do you mean leaf springs?
Their bonuses are they're incredibly cheap, they position the axle without wishbones, control arms, radius arms, and all the assorted bushings and hardware.
The leaves scraping together work as a crude shock absorber/dampener.
One can increase the load capacity by adding a leaf. Great for plow trucks where the design is altered radically with additional weight.
Disadvantages are one usually winds up with cheap solid axles with the cheap leaf springs, with a lot of unsprung mass. One wheel hitting a bump on one side effects the wheel on the other. If power is being transferred through that axle and one hits a pothole that whole axle will shake and bounce like crazy.
Springs work better, packaging wise, in the front since you can have them further outboard. The wheel can turn for steering through where the leaf springs would be. Macpherson struts and double wishbones, 99% of car front suspensions, use springs. Springs in the rear are all but standard now too and in my cars they're struts, ie springs wrapped around shock absorbers for better packaging. These work better for independent rear suspension which rides nicer.
So you'll find coil springs on nicer riding vehicles but there are other design elements at work.
You'll find leaves on utility vehicles but it's for their utility, flexibility, and economy.