In general, using green threads to run background tasks asynchronously will mean that your application can respond to the user before the mail is sent. You're not concerned about exploiting multiple CPUs; you're only concerned on off-loading the work onto a background process and returning a web page as soon as possible.
And from examining the Rails documentation, it looks like deliver_signup_notification will block long enough to get the mail queued (although I may be wrong). So using a thread here might make your application seem more responsive, depending on how your mailer is configured.
Unfortunately, it's not clear to me that deliver_signup_notification is necessarily thread-safe. I'd want to read the documentation carefully before relying on that.
Note also that you're making assumptions about the lifetime of a Rails process once a request has been served. Many Rails applications using DRb (or a similar tool) to offload these background tasks onto an entirely separate worker process. The easiest way to do this changes fairly often--see Google for a number of popular libraries.