Is there any reason to update Android firmware (5.1.1 -> 6.0.1) via PC using software like Odin, rather than just doing an OTA update? Is there any difference between the two methods in terms of performance, battery life, etc.?
Note: I've never found an incremental update that can be flashed via fastboot and I'm not familiar with Odin or the likes of it.
I can't give an objective view but note that there are multiple reports on this site where you would notice that OTA (it usually do a dirty flash) upgrade caused performance issues, such as sluggish animation, frequent app crashes (result of dirty flash), automatic reboot and issues with functionality of some apps. Often, a factory-reset fixes them.
In the case of flashing the OS from PC, you're most likely going to clear the data partition as well so the resultant, once the flashing is completed, would be equivalent of buying a device with OS shipped from a vendor. If the OS isn't faulty, you shouldn't be having major issues.
I speak from experience with OnePlus One. OTA never worked for me. They either caused overheating or radio issues.
It depends on how the full flashable tar/kdz was obtained. If it was obtained from an OTA capture, what difference does it make? Just doing the OTA would then be the most direct from the manufacturer. Capturing is a popular deal, because big companies don't do free hand outs. The more experienced (ex. Sammobile) actually build their own tar files that are independent flashables using a combination method (OTA + live system pulls). As anyone can see, it's all still based off the OTA.
Your question could then be a) whether wiping data after an OTA makes it "run smoother" and have better battery life. And that answer is definitely yes, because certain old files (like app data) is converted to run on the new system, but can cause problems and complicated interpretation. And b) if AOSP/vanilla/slim roms would be a better option, and that also depends. They will usually run better, but you may get more bugs/crashes.
The bottom line is you have to do your research, and probably rom testing. There is no one-size-fits-all answer.