I have the same problem. I use Google Apps as my email provider, GoDaddy as my host provider (Microsoft shared hosting account), I have set all the MX records in GoDaddy DNS setting to point to Google Apps.
When testing WP Mail SMTP plugin it wokrs like a charm. But I do not receive any email from the cotact form. I am using Woothemes Canvas contact page template as a form.
Any idea?