Both calibers are certainly more powerful than 5.56, which makes them a legal choice in some states.
The bigger differences are in smaller things. They use the same lowers, uppers, BCG, sights, triggers etc. In hardware, the barrel, bolt, and magazine are what is different, and you can get them from the same makers, to boot.
What it boils down to is what incremental increase in ballistic performance do you want? The focus of the 6.8SPC development was more lethality than 5.56 at combat ranges, from a 14.5" barrel. The 6.5G focus was more precision at long ranges on a competition range, using 20"+ barrels. The case design was optimized to promote these different goals, and is exactly why they are so different.
Long case, short bullets in a larger diameter carry more initial energy, and in the typical 16" carbine AR, offer a short, light handy package. Short fat case, long bullets in a smaller diameter carry slightly less energy,even from longer barrels, but the bullet carries it further due to it's aerodynamic efficiency. That starts showing results, but it's mostly beyond combat ranges.
The important factor is where you want the power, and the break is about 250-350 yards depending on load. I chose the 6.8SPC because ballistic efficiency doesn't contribute much inside 350 yards, which is a long shot hunting whitetail. The .30-30 didn't become the nation's #1 deer cartridge for 100 years knocking them down out beyond 300 yards, and I've never had that shot hunting in the last 30 years.
Since I wasn't going to carry a caliber that was set up for a small percent shot, I got out of .30 cal battle rifles and went to the AR in 6.8. It matches what most hunters shooting whitetails and hogs in edge environments actually need. Power under 350 yards in a small light carbine.
That's why it's the #1 alternate caliber and has dozens of suppliers. Just get the SPCII stuff, which is largely all that is offered. And you can buy ammo off the shelf, even if it is Rem 115 OTM at the local Academy Sports. What you can also buy is full power tactical loads, and for the price of a decent scope, enough equipment to make handloads. Those are capable of reaching 3,000 fps from a 16" barrel.
Bill Wilson of custom 1911 building has no problem knocking down hogs on his Texas property with the 6.8, and offers his for sale. Check Bison, White Oak, and ARPerformance for uppers, barrels, bolts, and accessories. The guys at 6.8forum have lots of experience and shoot BOTH calibers, you'll find straight answers there on it's performance. It's also easier to search, it's ALL 6.8.
Choose which you prefer, I built mine to hunt, and set it up for that.