The difference is you will be pushing a .284 diameter projectile with about the same amount of powder you would use for the .308. Both calibers will definitely do the job on either game. The only difference being is the .308 would have a better ballistic coefficient and will buck the wind a little better but I do not think you could even tell the difference. Ballistic ally the .284 is better because of its flatter trajectory and down range performance so if it's distance that your looking for the 7mm-08 would be the choice of most. If it's knock down power your looking for they both are adequate to get the job done however, I would choose the .308 myself. I feel it is really a personal preference thing.
Nothing like a 168 grain Barnes TTSX powered by 44.5 grains of Varget lite by a CCIBR 200 primer in a Norma case to put a hog down in his tracks!