It appears that this is a British/American distinction.
The Corpus of Contemporary American English (COCA) reports 92 incidences of “fill out a/the/this form” and just 2 of “fill in a/the/this form”, clearly establishing “fill out” as the standard idiom in American English.
I haven’t worked out how to search the British National Corpus yet, but I wouldn’t be surprised if the results were reversed there.
Edit:
OK, I got the BNC to respond to queries—although it sure takes its sweet time—and I got 19+7=26 results for “fill in a/the form” , and 5+1=6 for
“fill out a/the form” .
So it does appear that British English favors fill in over fill out , although not to the degree to which American English favors fill out over fill in .
english.stackexchange.com/users/39/nohat
Both are perfectly acceptable.
english.stackexchange.com/users/78/Alan Hogue
As an Englishman living in the US for almost 20 years, "fill out" still sounds jarring to my ears.
I had never heard it used before I came to America.
I rarely hear "fill in" on this side of the Atlantic.
english.stackexchange.com/users/858/Kevin Lawrence
In my dialect of American English, you "fill out the form" by "filling in the blanks" on the form.
P.S. to fill out the form is to complete it.
To fill in the form is to supply information as required.
english.stackexchange.com/users/94848/TRomano
The Americanism follows in line with other usages: Knock out (to complete, slang , verb )
Round out (add more in order to ensure a broad range, slang) fill out (no form-based, as in "
why don't you use the rest of those flowers to fill out this basket" )
english.stackexchange.com/users/23834/New Alexandria
Does one  fill out a form  or does one  fill in a form ? I've gotten different answers from the people I've asked.  Google search results:      fill  in  a form — 14,200,000  fill  out  a form — 7,000,000
"Fill out a form" or "fill in a form"
english.stackexchange.com
<word-choice><prepositions><phrasal-verbs><collocation><transatlantic-differences>
english.stackexchange.com/questions/1514
english.stackexchange.com/users/391/aviraldg
It appears the the difference between "fill in a form" and "fill out a form" may be a British/American distinction. It seems that British English favours "fill in" whilst American English favours "fill out". However, it is generally considered that both are perfectly acceptable. Both forms are acceptable, but "fill out" is much more common in US English, whereas "fill in" is more common (although less exclisuvely so) in British English.
It appears the the difference between "fill in a form" and "fill out a form" may be a British/American distinction. It seems that British English favours "fill in" whilst American English favours "fill out". However, it is generally considered that both are perfectly acceptable. Both are perfectly acceptable. This appears that this is a British/American distinction. British English favors  fill in  over  fill out , although not to the degree to which American English favors  fill out  over  fill in .
It appears that this is a British/American distinction. Both are perfectly acceptable. In American English, you "fill out the form" by "filling in the blanks" on the form. "Fill out" may sound jarring to English people. Whilst both appear to be acceptable, "Fill out" is more common in American English and "Fill in" is more common in British English.
It appears that this is a British/American distinction. Both are perfectly acceptable. As an Englishman living in the US for almost 20 years, "fill out" still sounds jarring to my ears. In my dialect of American English, you "fill out the form" by "filling in the blanks" on the form.  
This appears that this is a British/American distinction. British English favors  fill in  over  fill out , although not to the degree to which American English favors  fill out  over  fill in . Both are perfectly acceptable.
Whilst both appear to be acceptable, "Fill out" is more common in American English and "Fill in" is more common in British English.