Take postfix for example. It's an MTA, but is it also an SMTP server? Are all MTAs SMTP servers? Are any? Is there a difference between a mail transport agent and a mail transfer agent?
Can an MTA be configured to be an SMTP server, or vice versa? What's the difference between an MTA and an SMTP server?
An MTA is a part of an SMTP server; it is responsible for receiving emails from an MSA and then forwarding them to an MDA (if the destination of these e-mails is local) or to another SMTP server. Some people call MTA an SMTP server.
Something about flanker12x's answer doesn't seem quite right. Especially because wikipedia describes postfix as an MTA.
As suggested by @roaima, it's mostly okay to think of an SMTP server as an MTA that receives and sends email over the SMTP protocol.
MTA's that do not support SMTP are rare. So its hard to find an MTA that is not also an SMTP server.
MTA's usually have an SMTP server component:
It's acceptable to describe the single piece of software as both an MTA and an SMTP server. But if you are being really really pedantic, then the SMTP server is just the listening component of the MTA.
The two concepts can even exist separately...
Some software can have an SMTP server component without being an MTA. For example: issue tracking software or blog software might allow users to send email responses to threads. That software might have an SMTP server to receive the responses... But the email is immediately processed and discarded. It's not an MTA because it destructively processes the email.
This might be contentious but I'd say "no"; not necessarily.
Theoretically MTAs can transfer emails between each other by any protocol, as long as the email arrives. So an MTA is certainly allowed to accept email from other protocols. As long as an MTA can get an email to its destination, the protocol is irrelevant.
To demonstrate my point, Microsoft Exchange Server can send emails between accounts in the same organisation without using SMTP. It is still acting as an MTA in this context. Exchange Server does of course also have an SMTP server but it's not theoretically required.
Likewise postfix has multiple components defined in master.cf and the SMTP server can be disabled. As long as you replace it with something, postfix could remain defined as an MTA as long as the user agents can ultimately place email in its queues.
SMTP is already nearly 40 years old. We're unlikely to see big rise in MTAs without SMTP any time soon.
