When Testing SMTP Settings - what is the difference between doing at Site Settings and Host Settings?

When testing with a hostingpartners smtp server where does the test mail go - to the smtp server user or to the superuser/host logged on?

I have set up my dev environment to use smtp.gmail.com as smtp server. When testing from Host Settings/Advanced Settings I receive a test mail in my gmail inbox. However when testing the exact same settings on my hostingserver I get no email in my gmail inbox.

For further testing I have gotten an test account at the hostingpartners smtp server. When testing there is no reaction and I get no mail in my gmail inbox. The user I am logged on as is set up with my gmail-address.

What could be wrong?
What would be a way forward to solve this issue?