Is there any reason to update Android firmware via PC using software like Odin, rather than just doing an OTA update?

Yes.

• Since Android Lollipop, Google has moved to block based OTA updates which means if the system partition is tampered with (often done when rooting the device) the OTA upgrade would and should fail. How would you update the OS then?
• In Android versions prior to 5.0, OTA update still fails sometimes because checksum for a file in system partition doesn't match. How would you update the OS now?
• During OTA upgrade, some user reports the device being stuck in recovery mode. How would you update the OS now?

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.

Is there any difference between the two methods in terms of performance, battery life, etc.?

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.

Is there any reason to update Android firmware via PC using software like Odin, rather than just doing an OTA update?

Yes.

• Since Android Lollipop, Google has moved to block based OTA updates which means if the system partition is tampered with (often done when rooting the device) the OTA upgrade would and should fail. How would you update the OS then?
• In Android versions prior to 5.0, OTA update still fails sometimes because checksum for a file in system partition doesn't match. How would you update the OS now?
• During OTA upgrade, some users report the device being stuck in recovery mode. How would you update the OS now?

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.

Is there any difference between the two methods in terms of performance, battery life, etc.?

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.