As I said it's not strictly a matter of bullet size. If it were, the 30-30 would be on par with a .300 Mag as both offer .30 cal bullets of 180 (or 170) grains. There are several factors involved:
1. Bullet diameter (calibre). The larger diameter the larger the hole it makes, the more damage it does.
2. Bullet weight. This is high-school physics: Force = Velocity x Mass. Heavier objects have more force than lighter objects moving at the same speed.
3. Velocity. As was mentioned above; the faster the bullet is travelling, the more energy or force it has. This explains why some lighter calibres, particularly the .270 (27 cal) are more powerful than larger 30 caliber rounds (ie: 30-30, 300 Savage etc). Another advantage to higher velocity rounds is their trajectories are flatter for long range shooting; since the time of flight from firing to impact is less, the effect of gravity is minimized and they don't drop as far. Velocity is determined by many things such as bullet shape (aerodynamics), riflle barrel length but above all mostly it is a matter of how much propellent is in the cartridge.
In this case, at 200yds a 180 gr bullet from the .300 Dakota delivers apprx 3100 foot-pounds of energy on a target, whilst the larger calibred .308 can barely muster 2000lbs. At 300 yds and beyond, the difference is even more pronounced. (Taken from the ballistic tables in Petersen's 2007 Hunting Annual).
Further illustration, compare the 30 cal 30-30 vs the 25 calibre 25-06:
At 200 yds the 150 gr bullet from the 30-30 delivers barely 950lbs (900 lbs is used as a rule of thumb for the minimum force required to kill an adult deer), while the 25-06 has 1650; past 200yds the 30-30 is practically useless as a big game round while the 25-06 easily maintains over 1000ft/lbs well beyond 400 yds