I am an enthusiast and currently I shoot with Nikon D90. I am in the process of upgrading the body, but little confused over the same generic question of FX vs DX. I know that an FX sensor delivers better dynamic range, high ISO capacity and more control over depth of field than a DX sensor. However, if I shoot with a FX camera in DX mode does it deliver the same high ISO and dynamic range performance?
DX lenses project a smaller image circle than FX lenses, smaller than an FX sensor. DX mode allows you to use an area the size of a DX sensor in the center of your FX sensor so the DX lens image circle will cover this area. Since each photosite in this area is still its full FX size, you will retain all the ISO and dynamic range capabilities of an FX sensor. However, since you are only using about an area about 45% of the full FX sensor, you will lose about 55% of your megapixels. This means a 36MP FX camera will produce 16.2MP images in DX mode, and a 24MP FX camera will produce 10.5MP images in DX mode. In addition, in DX mode, you will experience the same crop factor effect on your focal length. This means a 35mm FX lens on an FX body will give the angle of view expected, but a 35mm DX lens on a FX body will give an angle of view approximately like that of a 50mm FX lens on an FX body, just as that lens would do on an DX body.
If you have a large investment in DX lenses, DX mode can be a reasonable bridge, but you will not be using about half of your very expensive pixels, so if you have fully committed to moving from DX to FX, you should seriously consider selling your DX and lenses and acquiring FX lenses to take full advantage of your sensor.
You're using the same sensor pixels, so in many ways, yes you will get the same high ISO performance and dynamic range - but that applies only for a pixel by pixel comparision. If you're looking at a fixed print size, which is generally a much more useful comparision, then you won't get the same performance because you'll have less downscaling going on (see Icycle's answer for details as to why).
What you will definitely lose is the better depth of field control of the FX sensor - as you're effectively reducing yourself to a DX sensor, you're back to the more limited depth of field control you get there.