Will the picture taken from
Not exactly, since the picture made on the DX sensor with the 35mm lens will have slightly greater depth of field for a given aperture, and the internal focus (which actually changes the focal length of a lens to achieve focus) may mean that the proportion of focal lengths at a given distance might not always be the same. But it'll be close enough for jazz, so to speak -- the main difference being that, wide open, the 50mm lens will be able to achieve a shallower depth of field. (You can always open up the 35 compared to the 50 at other apertures.)
35mm isn't exactly 50mm/1.5 (the crop factor), but then the stated focal lengths of lenses tend to be approximate rather than exact in any case. If the 50mm is slightly long or the 35mm is slightly short (or both), you may have an exact match. As far as I know, though, only Hasselblad makes a point of stating the actual infinity focus length alongside the design length.
If you compare a 35mm DX lens and a 35mm FX lens on a DX body you would see no theoretical difference.
If you put both lenses on an FX body, the centre of the resulting images would still be the same. The difference would be the corners of the image with the DX lens would be vignetted. Think of DX as being a cheaper design of an FX lens where the corners don't matter because they'll be chopped off anyway on the DX sensor. So the DX projected image doesn't extend out to the corners of an FX-sized sensor, but the middle part of the image is the same as with the FX lens.
The lens projects an image (you know, small and upside-down) onto the focal plane and that projection is a property of the lens and isn't be affected by what size sensor happens to there.
So a 50mm (DX or FX) lens will always project a different image than a 35mm (DX or FX).
The crop factor (magnification) comes from the sensor, not the lens. The same image projected from any lens onto a DX sensor, as compared to an FX sensor, will be magnified due to the sensor only capturing a centre portion of the image.
So to answer your question, on the same camera, the two lenses are absolutely different. On the same body, a 35mm is a 35mm whether it's an FX or DX lens. And a 50mm is a 50mm is a 50mm.
If you want to compare 50mm on an FX vs a 35mm on DX, then Stan has answered that, they are roughly the same, and I think the other questions reference above cover that as well.