Apart from the raw number of pixels, the ability to make practical use of the cheaper DX lenses, and a loss of apparent magnification with any given lens, you lose nothing.
Raw pixel numbers are only really meaningful if you customarily make very large prints (or do all of your composition by cropping in post). Even at that, the difference between, say, 12MP and 16MP isn't all that much, and there is some leeway in terms of pixels per inch that will create good prints; you'd need to step up to the 24+ MP class before you can really go places you can't go with 12MP.
You can use DX-format lenses on the FX (full-frame) cameras, but you're restricted to using the center of the sensor and only get about a 5MP image. That's good enough for the web, but not for a whole lot else. On the other hand, almost all of Nikon's high-quality lenses will cover the full-frame image.
The loss of magnification, though, can be a real issue, depending on the type of photography you do. If you rely on long lenses (for sports or wildlife, for instance), then the 1.5x crop factor of the DX (APS-C) cameras can save you a considerable bundle (with the penalty of a slightly greater depth of field, equivalent to stopping down one stop). On the other hand, if you tend to shoot wide, you get more bang for your buck with a full-frame camera.
That said, almost everything else is a plus. The D700 and, to an even greater extent, the D3 have much better handling from a pure photographer's point of view. There are no entry-level hand-holding features to speak of, but for someone who wants to take full control of the camera, everything is right to hand—there's not a lot of digging through menus to get where you're going; everything has dedicated buttons and knobs.
Because the sensel sites on the sensor are bigger, you get a wider real dynamic range and much smoother tonality (there isn't nearly as much quantization noise because the individual samples are much larger—the samples are statistically much more likely to represent the data stream, the impinging photons, more faithfully). You can count on getting a full stop better noise floor (for a given sensor generation; sensels that are twice as big are twice as likely to be struck by photons even in shadow areas). So even with a reduction in the number of pixels (compared to the D7000), you will get better image quality from a tonality standpoint.
The larger sensels also have another effect on image quality: lenses don't have to be quite as acutely sharp in order to render detail. A lens that would test as borderline mushy on a D7000 or D800 (which have approximately the same sensel density) can appear to be tack-sharp on a D3/D3S or a D700. As with the reduced quantization noise, this will give you better subjective image quality, and may be more important to your final image than the number of raw pixels you can capture.