This looks very much like the problem I'm wrestling with. I'm busy with developing a WordPress website and I have two test servers with WP 3.5 installed, one with a Dutch version and one with an English version. On the Dutch version I installed Contact Form 7 and WP-Mail-SMTP. On pressing the Submit button I receive the e-mails at the specified addresses. When I press the Submit button on the English version, I see the spinning arrows, going around and around for ever. I haven't the faintest idea what could be wrong. The Dutch version has been installed with a localhost in a Windows 7 system. The English version is running in a Linux server. I have verified the settings of the WP-Mail-SMTP plugin and I can successfully submit test mails via the plugin settings panel. I also tested after the installation of the WP SMTP plugin. The results were the same. On the plugin settings panel I was able to submit mails. On the webpage with the Contact Form: endlessly spinning arrows again.
I have the feeling that it looks like something missing in the communication between the two plugins. But I don't know if it has something to do with the usage of PHP running under Linux or a difference between the Dutch and English language versions of the plugins.