I have Samsung Galaxy Note 3. If I charge my phone using the same adapter but with different cables, does that affect the charging speed with the USB 2.0 cable or 3.0 cable, supposing the 2.0 cable is of best quality, so both cables are good themselves?
USB 2.0 cables and USB 3.0 cables don't charge your device any faster because: USB 3.0 only increases the speed of copying files to your PC and back, but does not affect the charging speed
To check the amount of current (in mAh) received by the mobile, use app called - CurrentWidget.
Only High Quality Cables will give you high current from the adapter. For example, when your adapter has max output of 2A and using high quality USB 2.0, you will get around 1900mA which is almost equal to Battery Output. On the other hand, when the cable is bent (due to frequent usage), the current transferred via the cable is less.
Also, USB 2.0 and 3.0 cable does not result in fast charge. Rather, USB 2.0/3.0 port on PC provides high current output as well as data transfer rate. On PC, USB 3.0 gives you around 2000mAh current with good cable, whereas 2.0 gives approximately 500mAh.
It won't make any difference , if you are charging from a wall adapter (assuming both cables are fine as mentioned)
But, if you are using your laptop, things change. USB 3.0 port compared to a USB 2.0 port, the former has a maximum rated capacity of 900mA, whereas latter has a rating of 500 mA (80 % faster). In this situation
USB cable 3 connected to laptop USB port 3.0 (your phone has USB 3 port ) will charge faster at 900 mA
USB cable 2 connected to laptop USB port 3.0 and your phone , which has USB 3 port will charge at 500 mA (since USB 3.0 port is backward compatible and can accept a USB 2 connection)
USB cable 2 or 3 connected to laptop USB port 2.0 and your phone , which has USB 3 port will charge at 500 mA
Also see related question from SE superuser Will a mobile phone charge faster if plugged into a USB 3.0 port
Note: Samsung Galaxy Note 3 (OP's) phone, is the first phone to have USB 3.0 port, subsequently discontinued in later models of Note series