Why does marginal cost (derivative of total cost) differ from variable cost at each level?

Why does the marginal cost equation (as the derivative of total cost equation) make predictions of variable costs that are very different from costs calculated using the Total Cost equation?

Marginal cost is simply the change in cost divided by the change in quantity.

MC = ΔC / ΔQ

However, marginal cost also can be computed using the derivative of the Total Cost function.

Suppose you have a short-term Total Cost equation for a production case in which no capital is used; labor is the only input.

TC = w * L

The production function is

Q = L^(1/3)   ... therefore
L = Q^3

And given that the w = 1, then

TC = Q^3

Therefore the Marginal Cost equation, as the derivative of the Total Cost equation, would be...

MC = 3Q^2

Of course, when Q = 0 then the TC equation and MC equation = 0. But raise Q to 1, and TC is now 1. Therefore, MC = ΔC / ΔQ = 1.

But the MC equation gives MC = 3.

The difference grows as Q increases more. When Q = 2, the TC equation returns 8, a cost change of 7, while the MC equation returns 12.

I understand mathematically why these are different, but I don't understand why both are supposedly correct and useful in economics.