We consider interactive communication performed over two types of noisy channels: binary error channels with noiseless feedback and binary erasure channels. In both cases, the noise model is adversarial. Assuming at most -fraction of the bits can be corrupted, we show coding schemes that simulate any alternating interactive protocol with rate . All our simulations are simple, randomized, and computationally efficient. The rates of our coding schemes stand in contrast to the interactive communication rates supported by random or adversarial error channels without feedback, for which the best known coding schemes achieve rates of and , respectively. As these rates are conjectured to be optimal, our result implies a large asymptotic gap between interactive communication rates over noisy channels with and without feedback. Such a gap has no equivalent in the standard one-way communication setting.