Different shutter speeds have an obvious different effect: more motion-blur. (That includes both subject motion and blur from any motion of the camera itself.)

Different apertures also produce different results; most notably that depth of field increases as you stop down. So, f/8 gives you a much deeper in-focus area than f/1.8. This effect is lessened on a smaller sensor (or, if for some reason you just crop out the center part of the image and blow it up), to the point where it's basically a non-factor on most point & shoot cameras.

But there are other effects of changing as well: lens sharpness, contrast, and vignetting characteristics change, usually improving significantly when you stop down a bit. (For a certain look, though, that technical improvement may not, in fact, be what you want.)

That's assuming perfect reciprocity — the idea that aperture and shutter speed really are perfectly interchangeable for exposure. With film that's not strictly true as you get to extremes — see this question on reciprocity failure. But for digital, it's not meaningfully a factor.