Sometimes changing the header is not enough, some sites check the referer as well:
curl -v \
-H 'Host: restapi.some-site.com' \
-H 'Connection: keep-alive' \
-H 'Accept: text/html,application/xhtml+xml,application/xml;q=0.9,*/*;q=0.8' \
-H 'Accept-Language: en-GB,en-US;q=0.8,en;q=0.6' \
-e localhost \
-A 'Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/29.0.1547.65 Safari/537.36' \
'http://restapi.some-site.com/getsomething?argument=value&argument2=value'
In this example the referer (-e or --referer in curl) is 'localhost'.