I had this problem too, using WordPress 3.8.1. WordPress plugin directory says this plugin is only compatible up to 3.6.1.
However, I found my problem could also be solved by altering the folder permissions AND ownership. In my case, I had to delete the /uploads/wpcf7_captcha that had been automatically created, and re-create it manually using FTP. I then set the permissions to 0777 and it works fine now.
Hope this helps someone.