I had the same problem but the noise ended up blowing a hole in the front diff case! I put a link to my post below. In summary, mine was grinding/humming then blew apart. Now it is dealer rebuilt (because GM doesn't offer full front diff assemblies) and hums under load. I work in the axle industry and the noise sounds like the hypoid gear adjustment is out of spec and it has improper tooth contact that is only heard under load. If this is the case, the bearings will fail prematurely. Anyway it is now back at the dealer and they don't know what to do. Frustrating.
http://www.gm-trucks.com/forums/topic/171347-2014-sierra-front-diff-issues/