Here’s something that you get with Fastmail that Gmail won’t let you do: forward emails without verification so that they work with services like ifttt.
I’d tried several work arounds between Gmail and ifttt to be able to email an attachment to email@example.com and have it redirected to firstname.lastname@example.org. Even though I could set up a email@example.com email, Gmail would force me to verify I was allowed to forward to this address. This process entailed having a message sent to firstname.lastname@example.org with a code or link I needed to click. I could set up a ifttt task to process incoming email as text documents, but the email would come from some random Google address that wouldn’t match the address my ifttt account used. I could never authenticate forwarding to email@example.com.
With Fastmail, you don’t need to authenticate. Under Options > Define Rules > Forward, you can set up rules to redirect emails to other addresses, such as firstname.lastname@example.org. Now, when I send an email with an attachment to email@example.com, the attached file will show up in Dropbox.