I know that there is snow outside; this knowledge is based in part on my visual experience. When I look out the window, I have experiences that partially constitute seeing snow. I also know that squares have four sides. Arguably, this knowledge is independent of experience, depending only on my conceptual competence, or rational capacities, or something like that. My knowledge that squares have four sides is not derived from experience, the way that my knowledge that there is snow outside is.
According to certain prominent rationalist views, what explains this difference is that my knowledge about squares, unlike my knowledge of snow, is based not on perception but on intuition. I have the intuition that p, and intuitions are a source of evidence, and so now I have justification for believing that p. It's hard for me to see how, on a view like this, the relevant knowledge comes out independent of experience. For intuitions are experiences, every bit as much as perceptual experiences are. Indeed, some rationalists characterize intuitions phenomenologically: intuitions feel a certain way, and having that sort of feeling provides justification for intuitive beliefs.
On this sort of view, intuitions look to be just another kind of way of experiencing the world. One way that we experience the world is by seeing things; another is by intuiting things. You can call things learned that latter way a priori if you want to, but this just doesn't look to me like belief independent from experience. On the contrary, on this sort of a view, it looks like the a priori beliefs are those that are based on a certain kind of experience: the intuitions.