The Fan Only connectors on the PSU are optional fan outputs, which are equipped with two pins in a four pin drive power connector and equipped with 24ga. leads. They are marked Fan Only, because they cannot power a drive.
You can use either the motherboard, or the PSU to power your fans.
If your chassis fan has a tachometer output (i.e. three wire) connector, then you should connect it to the motherboard output. This way the fan can be checked by the BIOS and you can set-up your BIOS to detect a failure during POST. If you're exceptionally anal, you can also use a real time fan software monitor with this arrangement.
If your chassis fan uses a two wire with four pin drive power connector, then it's simpler to connect it to the PSU fan output.
Note: you should never power a fan that draws more than 2A using a fan output. Common fans use less than 0.5A, but if you're building a combination vacuum cleaner/hovercraft/air cooled PC then you should connect those immense rotodyns to 18ga. wired drive power connectors.