For birds and most wildlife, reach matters more than f/2.8. On a Canon 60D, 200mm is usually too short, and even 300mm is often still limiting for frame-filling bird shots. So between your two choices, the 70-300mm is the more sensible wildlife option.

That said, several photographers would still find both lenses frustrating if wildlife is the main goal. A lens reaching 400mm or more is commonly preferred for birds. Suggested alternatives from the discussion were the Canon 100-400L and Sigma 50-500mm. If you want a lens that is also excellent for portraits/events, a Canon 70-200mm f/4L IS was also recommended for its optics, but it still won’t solve the reach issue for birding.

If you do choose the 70-200mm f/2.8, its advantages are versatility and speed for low light, concerts, portraits, and possible use with a teleconverter. But strictly for wildlife, longer focal length wins.

Also remember that technique matters: blinds, feeders, water, and predictable perches can help you get much closer to birds than lens choice alone.