- As already mentioned - less depth of field, if you want to / need it. This wont help if you want the opposite, though.
- In general a larger sensor also translates to more light collected, and larger pixels, thus better low ISO performance. This is actually a very noticeable difference (of about a stop or two) in compareable models.
- Obviously also higher resolutions (D800 vs D5200).
- The larger your sensor, the lower the demands on your optics for the same result. For example, there are a lot of discussions about what lenses to use with the Nikon D800 and D800E. However, the cameras themselves only have 15 Megapixel resolution if used in DX mode. So the newest DX cameras D3200 and D5200 with 24 Megapixel resolutions actually have a higher per area resolution (and a smaller pixel size) than the "resolution flagship" D800. This translates into even higher demands on the optics. Fortunately, D3200 and D5200 are usually not bought by people with high demands on the pixel quality of their images, so while D800E owners are thinking about which lenses to use best, users of D3200 and D5200 users are much less critical of their lenses.
- Specifically for Nikon, the DX lens offerings are lacking, and Nikon doesnt seem to be interested in changing this any soon. For example, many people want a 24mm f/1.8 DX lens. You can of course use FX lenses as well, but they are larger and heavier and more expensive than necessary, and they have different meaning thanks to the crop factor. Thats why many people just get the 50mm f1.8 FX, which is actually an 76mm equivalent on DX, and thats it. Getting a 24mm f1.4 FX to have a 35mm equivalent on DX, for example, would be extremely expensive.
- In the long run, if sensor costs continue to drop, the full frame format aka FX will probably completely replace APS-C aka DX.