In both cases, there's a low-dimensional latent space, and there's a pixel space of images. In the case of Eigenfaces, the correspondence between the two is linear. That's not a bad approximation, but "make him smile" or "make him bold" is obviously a very nonlinear operation. Eigenfaces approximates it with something linear, and that's just not very good. The justification for all the added nonlinearity in the mapping is that this operation really becomes linear in the latent space.