IMHO, shot placement is the most important factor for self-defense. I don't know much about the 6.8mm round so I won't comment about its performance. A larger diameter bullet will improve your chances of severing major blood vessels and cause more permanent damage to tissue and organs. Notice I said permanent damage, and not temporary. Many higher velocity bullets produce a large temporary cavity that expands the affected area. This tissue is not destroyed, only temporary shoved aside.
The more damage you permanently destroy, the more effective the round. Bottom line, shot placement is the most important factor. A .22 LR to the cerebral cortex will be much more effective than a .30 cal to the foot.
The 5.56x45 is a great round because it is easily placed on target and many more rounds can be carried because of the cartridge's profile and lighter weight. The downside is the bullet's small size and light weight. This is offset however, because of the round's ability to be placed well and placed well when the shooter is under stress, making it very manageable.
When considering a round you should choose on the largest bullet, both diameter and weight, that is the most manageable for you. If possible, you should shoot both before you decide. But then again, why not buy both?
Edited for typographical errors.