Why are heavy things harder to lift?

According to multiple websites, any object in free-fall (no air resistance) on earth will accelerate towards the Earth at 9.8 m/s. If all objects fall towards the Earth at the same rate, regardless of their mass, then why is it harder to push a heavier object away from the Earth, than a lighter object?

• Note that both current answers contain the same (small) error as this question: the acceleration is 9.8 m/s^2. m/s is the unit of speed, m/s^2 is the unit of acceleration. – Oebele Feb 23 '15 at 9:04

When you are lifting an object, you are exerting a force that balances the force of gravity on the object. By $$F = m g$$ where g is the acceleration due to gravity, you see that a greater mass causes a greater gravitational force that has to be balanced by the force you apply to the object by holding it or lifting it at a constant velocity.
Using the more general Newtonian law of gravitation, $$F = G \frac{M m}{r^2}$$ with $G$ being the gravitational constant, $r$ the separation and $M$ and $m$ two masses, we can rearrange for an object in the gravitational pull of Earth: $$F = G \frac{M_{Earth}}{r^2} m$$
If we approximate this by saying that the object is very close to Earth and its mass is nearly zero, the first part of the formula depends only on Earth's physical properties and therefore becomes a constant: $$g = G \frac{M_{Earth}}{r^2}$$
We see that $$g = 6.67 \times 10^{-11} \frac{5.97 \times 10^{24}}{(6.37\times 10^6)^2} \frac{N m^2}{kg^2} \frac{kg}{m^2} = 9.81 \frac{N}{kg} = 9.81 \frac{m}{s}$$
The acceleration of any object due to gravity is $g = 9.8m/s$ and this constant does not depend on the mass of the object (or the speed of the object or anything else for that matter, as long as you're on/near earth). Pushing an object away from the earth is another story. While the acceleration of objects towards earth does not depend on their masses, their weight does, $Weight=F=mg$. Two balls, one made of steel and one made of cotton, falling towards the ground will be accelerating at the same rate, but they will both hit on your head with different effects. In short, it's harder to push ($W=F=mg$) a heavier object because it has a larger mass, but acceleration is always just $g=9.8m/s$. The acceleration, together with the mass, causes weight.