The relationship between these two things is that greenwashing is one form of public relations.
Public relations can be defined as communications between a company and the public that are meant to project a certain image of the firm to the public. Greenwashing is one form of this. It is a form in which the firm tries to use public relations to project the image that they are an environmentally-friendly firm. To critics, this is merely a facade that masks practices that are not truly "green."
Greenwashing, then, is a somewhat deceptive form of public relations that is meant to make a firm seem more "green" than it truly is.