I grabbed the update from Kies - it's genuine ICS. If you go into Settings and then About Phone after the update, it reads that it is Android version 4.0.3. Plus there are new things that didn't come on the original 2.3.6, such as a very different Sammy app store (don't download from them - a lot of their apps contain embedded ads - the kind you have to opt out of or it'll show stars on your status bar) and the settings boards look very different, among other things.
Now - for the root part. I had the phone rooted before the changeover, but once that took place it was bye bye root. I used the same method I did for the initial root, only now you need to use a different version of the superuser.apk in order to make it work. The one you used for the pre-ICS update root WILL NOT WORK. It took me almost an hour and traversing quite a few places on the Web before finding the new superuser.apk for ICS, and unfortunately I don't remember where I got it from right now. All I remember is that I downloaded it onto my PC and copied it to the SD card, then flashed it on the phone. If I can find where I got it from I will definitely set up a link.
WELCOME TO THE HOUSE HALFWAY UP THE NEXT BLOCK!
CURRENT PHONE: Verizon Galaxy Note 3 (rooted)
CURRENT ROM: MODIFIED STOCK
A GOOD NANDROID BACKUP CAN SAVE YOUR A**, BUT IT CAN'T DO S**T IF YOU CAN'T ACCESS IT.
BE CAREFUL WHEN FLASHING A ROM, MAKE SURE IT'S MADE FOR YOUR PHONE AND FOLLOW THE DEV'S INSTALLATION DIRECTIONS
TO THE LETTER!!!!