You can use the same powder in the two rounds, but for best performance you'll find you need to pick one that favors heavier bullets in the AR if it works from middle to heavy in the .308, and vice versa. AR-Comp can be used over a wide range in both cartridges, but favors heavier bullets in both chamberings. RL 10X is good for lighter bullets in both. But the middling powders, like H4895 or Varget, I think you'll find run the .308 fine from about 150 to 180 grains, but in the AR just won't be best performers with bullets under 62 grains.
But YMMV. All guns are individuals. I would generalize that if you want to shoot light bullets in the .223 and heavy bullets in the .308, you'll probably end up with two powders. Your barrel twist rates may make this decision for you, though. .308's are usually 10" or 12" twists, and those handle the normal weight bullets. But .223's come with anything from 7" to 12" twist commonly, and few 14" barrels are out there. So, say what you have and then well know better what you can shoot.