The CV photograph is read slowly, which changes everything
BetterPicA profile picture is scanned in a list at small size. A CV photograph is looked at, in a document, next to a block of text, by someone who has already decided to give you thirty seconds.
Opposite conditions, opposite requirements.
What that implies
Formality is the safe default here, which is not true anywhere else. A formal crop, a neutral background, nothing competing with the text beside it.
The failure mode is looking careless. That is different from the profile picture's failure mode, which is being illegible.
Why one image rarely serves both
A crop that reads as relaxed in a document becomes a small head in a large frame at profile size. A crop tight enough for a list looks severe on a page.
Two crops from the same frame solves it and costs nothing.
The question that comes first
Whether to include a headshot at all. That is geographic rather than aesthetic — standard in much of continental Europe, actively unusual in the UK and US, and often stripped before shortlisting.