This forum is about Samsung Note 3 and it is NOT about clones or fakes. Asking for support on clones and fakes in any of the Samsung Note 3 forums is unacceptable!
Asking how to spot the differences between fakes/clones and "the real thing" is however acceptable, but given the fact that we now see 2-3 posts like that every day it is really becoming a major nuisance, so let's keep all this in a single place and avoid repeating the exact same questions again and again.
There is also a Note 3 Neo (N750/N7505) which is somehow similar in design to the Note 3 but with only 2 GB RAM and 1280*720 screen - there is a separate forum for those models here!
PLEASE DO NOT START ANY OTHER THREAD ASKING FOR THE SAME THING AGAIN AND AGAIN!
If you have new information relevant or new questions please post under this thread!
Currently there are many REAL Note 3 devices from Samsung, including (but not necessarily limited to) SM-N900, SM-N900(various letters), SM-N9000(various letters), SM-N9002, SM-N9005, SM-N9006, SM-N9008, SM-N9009. So far the overwhelming majority of the original devices have 32GB internal flash but there seem to be some 16GB devices listed/sold by reputable sellers in the world, so that will not be a good test. Please also note that (as I write this) there are 4 OTHER official Note 3 forums on xda - see links in the table-like list below.
Samsung Exynos models MODEL NAME / MARKET XDA/LINK N900 Exynos Version International HERE N9000Q Exynos Version Turkey HERE Qualcomm Snapdragon models MODEL NAME / MARKET XDA/LINK 2G | 3G | 4G (LTE,MHz) N9005 International LTE HERE 850/900/1800/1900 EU N9005ZWE SamsungUK 850/900/1900/2100 HK N9005ZKA SamsungHK 800/850/900/1800/2100/2600 N900W8 Americas non-US LTE HERE 850/900/1800/1900 Canada, Mexic 850/1900/2100 ('happy' bootloader) 700/1900/2100 N900T US T-Mobile LTE FORUM 850/900/1800/1900 850/1700/1900/2100 700/1700/2100 N900P US Sprint 4G FORUM 850/900/1800/1900 1900/2100 CDMA800/CDMA1900, LTE1900 N900A US AT&T LTE FORUM 850/900/1800/1900 (locked bootloader) 850/900/1900/2100 700/850/1700/1900/2100 N900V US Verizon 4G FORUM 850/900/1800/1900 (locked bootloader) 850/900/1900/2100 CDMA800/1900, LTE700/1700/2100 N900S South Korea SK Telecom THREAD 850/900/1800/1900 850/900/1900/2100 700/850/1800 N900K South Korea KT Telecom THREAD 850/900/1800/1900 ? 2100 ? 900/1800 N900S South Korea LGU+ THREAD 850/900/1800/1900 ? ? N900R4 US Cellular ? 850/900/1800/1900 850/900/1900/2100 CDMA800/1900, LTE700/850/1700/1900/2100 N9002 China Thread 850/900/1800/1900 850/900/1900/2100 NO 4G but 2nd SIM N9006 China dual-SIM ? Thread 850/900/1800/1900 850/900/1900/2100 NO 4G but 2nd SIM N9009 CDMA ? ? ?
The exact same letters CAN be found on some of the clones, so that is also not a good/definitive test.
Also some of the information displayed by system-information programs CAN BE FAKED - there are now clones that report 1920x1080 screens and 3GB RAM (which in itself should be a red flag since for instance an original N9005 should report around 2.38GB in RAM Manager, the rest being mapped to video/3D).
At this moment the best way to check if a phone is a clone/fake seems to be the SPen - the original Samsung pen is a device (similar/compatible with certain Wacom pens/digitizers) that is wireless-powered from about 10-20 mm away from screen of an original Note and on the home screen from about 10mm away the special pen hardware will provide a position (without any contact) and you will see a pointer and eventually some "hoover" actions; you can also press the special button on the side and you will get the special circular menu (aircommand). So far no clone has such a digitizer/pen. (the aircommand when you REMOVE the pen from the Note3 CAN be easily faked, since that one is another sensor that only feels when the pen was removed).
The other dead giveaway is the actual screen and screen resolution - some of the clones also fake the reported screen resolution as 1920x1080 but on the fakes when you take a screenshot (with a palm gesture or with Home+Power) that one does NOT come as 1920x1080 but instead as something like 1280x720 (other known clones have 960x540 or so).
Also if you are accustomed to the Samsung AMOLED screens you can immediately see the difference to the LCD or even IPS screens - outdoor is harder but indoor if you force the brightness from lowest to highest you immediately see the backlight on LCD/IPS screens and the pure black on the AMOLED.
Please also note that no Samsung Note 3 has any Mediatek processor like MTK6589 , MT6572 , MTK6572 or similar, so all those are fakes.
Apparently none of the clones (so far) have NFC and/or IR. Also the battery is smaller.
Also a very good test seems to be the presence of Odin/Download mode (from full power off with Volume Down + Home + Power) and Recovery mode (from full power off with Volume Up + Home + Power).
Here are some threads on some of those clones/fakes (inside you can see how hard is to tell just by looking at the device, and you might also find extra specific support if you have one of those devices):
Somewhere around there are even videos with some of those devices.
There are already some other excellent posts added to this thread, thank you very much to all the users that have contributed!
I feel the need to add that most advice so far is for the cases when you can test the device in your hands, in case that is not possible please just use your good judgement and if it looks too good to be true chances are that it is a fake.
I have also seen another difference that you might be able to spot before opening the original box - the Samsung box has at least 2 (most often now 3 with the one on region-lock) stickers/seals applied in such a way that you can not open the box without cutting those. One of them on Samsung phones also has a copy of the IMEI - which I believe for most real Samsung phones starts with 35 or so. None of the fakes that I have seen (on videos) have any of those.
Thanks to user sap7e here we also have a picture with some of the more visible differences!