Google Apps email versus GoDaddy email. Is there even a comparison here?
Your website is not hosted with Google so you need to change the MX record of any "local" mail exchanges so that, for example, a form that goes out from the domain does actually go out to the mail exchanges at Google's servers and doesn't get trapped in a local loop.
If your website and email are both hosted in the same place, you don't need to get into these details, but we're hosting our website at a host and we're hosting our email with Google Apps. Here's how to change your MX records so Google Apps hosts your email.