Jim is right that an apples-to-apples comparison of FX to DX means keeping the same distance and changing the lens, but other examples will show where the DX format hits the wall. Say, for example, that you are shooting a portrait with an FX body at 85mm, f/1.4 and a distance to subject of 4 feet. You're getting that nice, razor thin DOF of 0.05 feet. (You know, where the near eye is in sharp focus and the far eye is slightly OOF.)
Now you pick up your DX camera to get exactly the same shot. Well, not exactly because you would need a 56.7-mm lens, but let's call 50 mm close enough since that's all we have available. Now, I still want that 0.05-foot DOF, so let's just open up the aperture on that 50-mm lens. Let's see... according to the DOF calculator I need an f-stop of... something wider than f/1.0. Whoops!
Of course, if that kind of extremely narrow DOF isn't something you shoot, the ability of FX to get there isn't important to you. But if it is important to you, DX isn't an option.
Jim is also right that what DX may taketh away with respect to diffraction, it giveth back in using a wider aperture for a particular DOF. Example: You are shooting a landscape using FX at 24mm with a foreground subject 4 feet away, and you want the DOF to extend from 4 feet to infinity. You select an aperture of f/8 to get a hyperfocal distance of 8 feet. Switching to DX, we find that the 18-mm lens that gives us the same FOV can achieve that hyperfocal distance at f/6.7, reducing diffraction.
But it's not a one-to-one tradeoff. In this example, if both sensors are 24-megapixel, the FX sensor's diffraction limit is f/11.3. That's about 1-1/8 stops below the f/8 we are shooting at, and we are unlikely to suffer diffraction effects. But the DX sensor's diffraction limit is f/7.4, less than 1/3 stop from the f/6.7 we are using, and diffraction effects are likely to be creeping into visibility when the image is viewed at 100% or printed very large.
The bottom line is that FX has some advantages at the margins. Whether those marginal cases are important and worth the price of FX, only you can decide.