1. The problem is to subtract the repeating decimals $0.6\overline{6}$ and $0.3\overline{3}$.\n\n2. Recall that a repeating decimal like $0.6\overline{6}$ can be expressed as a fraction. For example, $0.6\overline{6} = \frac{2}{3}$ and $0.3\overline{3} = \frac{1}{3}$.\n\n3. Using this, we rewrite the subtraction as $$\frac{2}{3} - \frac{1}{3}.$$\n\n4. Since the denominators are the same, subtract the numerators: $$\frac{2 - 1}{3} = \frac{1}{3}.$$\n\n5. Therefore, the result of $0.6\overline{6} - 0.3\overline{3}$ is $\frac{1}{3}$ or $0.3\overline{3}$.