What is the difference between average cost and marginal cost? $$AC(q) = \frac {C(q)}{q}$$
$$MC(q) = \frac{\partial C(q)}{\partial q}$$

These are the definitions. But I don't understand what the difference is. 

Is the average cost the cost per unit while the marginal cost is about infinitesimals?
The average cost is the cost on average: total costs (C) divided by total number of units of production (q). Just as the equation you gave, says

The marginal cost is the cost at the margin: the additional cost of one extra unit of production, just as the equation you gave, says. Most goods and services come in discrete units: in those cases, it is not about infinitesimals, it is about the last unit of production, the unit that was produced to meet demand, the unit that would not have been produced had demand been one unit lower. In cases of continuous supply, the marginal cost is the gradient of the curve of total cost plotted with respect to total supply.

In pretty much all real-world circumstances, marginal cost is generally greater than average cost: that's guaranteed by a strictly monotonically increasing supply curve (with the trivial exception of the first unit of production, where average cost is by definition equal to marginal cost).