The idea of capture argues that regulatory agencies will invariably get "captured" by the industries that they regulate. This says that industries will be the ones that care most about the regulations that the agencies make. Therefore, they will try hardest to affect those regulations. In addition, the regulators will be concerned with trying to make the industries happy because A) they have to deal with the industries all the time and B) the industries often have influence on the politicians who control the agencies.
By contrast, the public interest theory argues more that regulation will actually work the way that it is supposed to. Bureaucrats and politicians will make and enforce rules that will protect the public. They will do so without being unduly influenced by the industries that they are supposed to be regulating.
So I guess I would say that the major difference is that the capture theory says agencies will be controlled by the industries they regulate while the public interest theory says the agencies will actually regulate effectively.