They fill about the same niche. I about bought a 6.5G upper from AA, then I looked at the ammo and parts availability and switched to the 6.8. Bill Alexander has eased up recently, but for a while, he held the specs and manufacturing rights so closely that the cartridge could not grow as fast as it otherwise might have.
6.8 was the opposite as far as its non-proprietary nature - everyone and their brother could produce ammo and barrels. The first 6.8 ammo (Remington) was too weak, and the first 6.8 barrels were too fast (twist) and the chambers were too tight because makers used .270 Win blanks. A couple of early-adopters did lots of testing and learned that the ideal twist rate for the 6.8 is 1:11 or 1:12, and the chamber needs to be SPCII or DMR, which are similar to the .223 Wylde (i.e. between 5.56 and .223).
You can fire a little heavier bullet with the 6.5G, but you can still get a 130gr pill with the 6.8.
Silver State Armory is the primary ammo supplier for 6.8 and they've got a new plinking round due out this spring. I hope it's about $14/20. It's not .223 cheap, but I'm not sure we'll ever see "cheap" ammo again. SSA also has premium hunting ammo with Barnes or Accubond bullets. The OTM bullets are advertised as LE ammo, and they load a TNT bullet for varmints.
The 6.5G needs a little longer barrel. I think they say 19.5 is the right balance between length and velocity loss. Of course longer is always faster, but you gain less per inch above that length.
The 6.8 was purpose-built from the start for shorter barrels, so "long" for the 6.8 is 18" and many guys run SBR barrels as short as 11", with most guys running 16".
When you compare ballistics, don't look at the 6.8 SPC ballistics provided by AA. They use the worst-case, weakest ammo to make the 6.5 look better. Of course it's just marketing and I in no way fault them for that. Here's an enormous reloading database of 6.8 bullets, powder charges, and MV to help your comparisons. The 6.5 crowd will always have the BC trump card - you have to decide how much that matters to you.
http://www.yuntaa.com/FileManager/Download.aspx?ContentID=4839637F63FAAB4CE040A8C0030270ED
http://68forums.com/viewtopic.php?f=9&t=1380