I'm having the same problem in a client`s website. I run a few wordpress' websites using CF7 and just one of them seems to have problems.
After installing the SMTP plugin I could send the test mail, but the CF7 still doesn't work.
I guess it`s some issue with the theme, does anyone here use the Modernize theme from themeforest ( http://themeforest.net/item/modernize-flexibility-of-wordpress/1264247 ) ?