When referring to a form,
In the US they are both used interchangeably, whereas in the UK we tend to say "fill in".
"Could you please fill in this form whilst you wait?"
"Due to COVID regulations, you're required to fill out this form"
Either of these would be ok.
You can get around this by saying
"Please complete the form below"
However when referring to a single field in a form, "fill in" would be better
"Please fill in your name in the box below"