The probability that Phillip flips $ k $ heads is $$\binom8k\left(\frac23\right)^k\left(\frac13\right)^{8-k}=\frac1{3^8}\binom8k2^k,$$ since there are $ \binom{8}{k} $ ways that $ k $ out of $ 8 $ coins will come up heads, and each of these arrangements of $ k $ heads out of $ 8 $ coins occurs with probability $ \left(\frac23\right)^k\left(\frac13\right)^{8-k} $. Therefore, the ratio of the two probabilities in the problem is equal to $$\frac{\binom832^3}{\binom822^2}=\frac{8\cdot7\cdot6}{3\cdot2\cdot1}\cdot\frac{2\cdot1}{8\cdot7}\cdot\frac{2^3}{2^2}=\frac{6}{3}\cdot2=\boxed{4}.$$