Humm.
Battery is way better on the Razr HD. Either 2550 or 3,300 vs. 2100 on the S3. However, it is not removable on the HD. But with 3,300 you "shouldn't" need to.
Screen. Same technology AMOLED. HD is 4.7 with soft on-screen keys that usually take up about .3 of that display area unless the app says no like movies. Movies will be 4.7 with no buttons shown. The S3 is 4.85 with button on the phone, not the screen so you always have the 4.85 for viewing. Big plus for the S3.
I had the S3 and the quality was fine as was the looks. However, the Moto phones are supposed to be much better quality. I believe that to be true. Looks are subjective. I like the HD over the S3. Some really hate the S3 and some really hate the HD. Your call.
S3 has 2gb of ram for memory and running programs (not storage). The HD has 1gb. SHouldn't be much if any difference, but I'd prefer the 2gb easily.
Cell phone reception. This is where the HD should rock the S3. Many, but not all, have had poor reception on the S3. I returned it for poor reception compared to my DX and my wifes Razr Maxx. If you live in an area with a weak signal, might have issues on the S3. Moto "should" be better reception.
Same processors.
Memory S3 has either 16/32/64gb. HD has 16/32gb. Both have memory card slots so that shouldn't matter.
Screen quality. Subjective. I tested 4 S3 and all had dim screens. That is the way they are, AMOLED. The current Razr AMOLED is a little brighter than the S3 side by side. If they improve the current razr display to the new HD (larger and hiDef), I'd suspect the same brightness or more. If so, the HD should be brighter than the S3.
Reports say the HD with color boost (marketing hype) is pretty vivid.
Camera: The S3 will "probably" have a better camera. Both are 8mp, but that isn't everything. The S3 takes pretty good camera phone pictures. Moto has typically been average. Definitely not a leader in this area. The camera, supposedly the same as the current Razr is just acceptable/average. Don't expect much from the HD. I carry a camera most the time, the average phone camera is great for those times when I don't have the dedicated.
Size. What ever you are comfortable with. The HD Maxx is thicker, but packs a much larger battery. The S3 can accept a new higher capacity battery that will make the phone thicker and thus will be fatter than the HD Maxx.
Moto typically has better accessories. My Moto car dock and desktop dock are sweet. I havent seen a good car dock for the S3, but by now they probably exist. S3 also has a big choice of nice desktop docks. Wash between the two. I use my moto as gps in my window. The power port on the side makes for a better car dock. S3 needs to plug into the bottom of the phone and when in landscape for gps, now plugs into the side of the dock. Still works fine though. I use the desktop dock also which mounts the same way, landscape. S3 docks will be portrait mode. I like landscape.
My opinion: Moto is better quality with better reception and way better battery 3,300 vs. 2100. I can use the aux head phone jack and bluetooth at the same time. I have my aux cable to my car stereo and when a call comes in my car takes it over BT.
S3: Larger screen is sweet, but dim. Poor reception on mine. Cannot use the headphone jack to put music to my car and BT at the same time. I have to unplug the head phone jack to get the voice to come over the bluetooth.
I returned my S3:
1) Dim screen
2) Poor call reception
3) No aux and BT at the same time. (trivial, but I use both simultaneously daily)