Nikon 24-120mm f4 or 16-85mm on D5100?

HI, I've been photographing with my new DSLR for a few months. I have a D5100 and was wondering which lens is better. I know the 24-120mm is a lot more but price isn't a concern. I know the 16-85mm is an FX lens and the D5100 is a DX body. Which one is better overall in terms of picture quality and build quality.

Added (1). Yea, 16-85mm is a DX lens, typo

Seems like you've got things reversed and confused. Both the D5100 and 16-85mm are DX format. Of all the items you've mentioned, only the 24-120mm zoom is FX format. The 16-85mm is a better option for most people who shoot the D5100 as it covers both wide angles and telelphoto with a field of view that is very similar to that of the 24-120mm zoom on an FX format body like the D3 series or D700 (not to be confused with the D7000).

In terms of image quality, for the focal lengths and apertures the two lenses have in common, the difference is negligible. You could certainly measure the difference in a lab but, in normal use, there's no way you could look at a photo and tell which lens was used to capture it without seeing the EXIF data attached to the file. Their optical formulas are actually quite similar with respect to the number of ED and aspheric elements in each lens. Two minor advantages of the 24-120mm are the new "nano" lens coating and its 9-blade aperture but, that's negated by the 16-85mm zoom's ability cover wider angles for indoor groups or landscape/architecture.

Where build quality is concerned, the 16-85mm is a lighter and smaller lens. On a very small, lightweight body like the D5100, that is advantageous to me as a general purpose lens. A heavier lens like the 24-120mm on such a small body is always going to make the camera feel like it wants to flip out of your hands or is otherwise simply nose heavy. Filters and polarizers for the 16-85mm are also cheaper since you only need a 67mm filter as opposed to 77mm filters on the 24-120mm. The point lurking here is that there are some tradeoffs between these lenses. And while price may not be a factor, the more expensive lens isn't necessarily the best lens for your application.