There are different generations of HK416, so it gets complicated depending on what's being compared to the MR556 (of which there are 2).
Most of the differences are pretty obvious when comparing product photos on HK's website.
http://hk-usa.com/hk-models/mr556a1/
http://www.heckler-koch.com/en/products/military/assault-rifles/hk416-a5/hk416-a5-11/overview.html
The current MR556A1 ships with a keymod-ish (HKey MRS) fore end. The older 556 used a quad rail identical to what had shipped on many 416 models (up to the A3). The current 416 A5 has a FDE quad rail with an integrated folding front sight.
The barrels and gas blocks are different. MR556 barrels are profiled differently (heavier) and not chrome lined. MR556 gas blocks are not vented and none have sling loops. HK416 A3 10" gas blocks are vented and have sling loops (all until the A5 had the loops). The HK416 A5 has an adjustable gas block. The HK416N (for Norway) was the first to feature an adjustable gas block.
You can see some differences in the buttstocks and castle nuts.The HK416 A5 has ambi lower receiver controls, while the MR556A1 does not (neither did HK416 up to A3).
You'll find original HK416 uppers (A3 and earlier) for sale if you look. More common are MR556 uppers modified to resemble them. Often people will swap a genuine 416 gas block onto their MR556 barrel. Sometimes they use the 556 gas block and drill a vent hole when shortening to 10". This is common when combined with a Geissele rail, because the sling loops are not compatible (ground off when mounting on real 416). If the barrel isn't chrome lined, then it came from a 556. The uppers are laser engraved with the model, but could be re-marked.
The benefit of the Geissele and Remington rails are they are lighter, longer, and more modular than the original HK quad rail. Word got around years ago that Delta/CAG was using the Geissele rail on their 416 10" and DEVGRU/ST6 was using the newer Remington Defense rail on theirs, so this makes them very popular with milspec cloners.