If we were right at the "edge"--which is very difficult to define, as no galaxy has a sharp boundary, but let's say another 12,000 light-years--the differences would be real but relatively subtle. The Milky Way would indeed be brighter in one direction, and nearly nonexistent in another, but it would still cover such an enormous area of sky that it wouldn't be especially dramatic. The lower rate of star formation in the outer Galaxy probably means that there would be fewer bright stars (which are typically young and massive) and nebulae visible than there are to us, so the night sky would be correspondingly duller. Interstellar dust would still visually obscure the inner regions of our Galaxy, perhaps even more than it does from Earth due to the greater distance.
What does change things is moving the Sun out of the plane of the Milky Way itself; if we moved the Sun and say, lifted her several thousand light-years above the galactic plane we would get an oblique, relatively unobstructed view of the entire disk, spiral structure and all.
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