They are really close! You can get larger bullets for the .308. If that is a concern.
Overall the 7MM-08 will be a little flatter shooting than the .308. The 7MM-08 also will retain its energy a little further down range than the .308, and recoil is a tad bit tamer with the 7MM also.
I doubt that any animal would know the difference in being hit with either cartridge though!