A "full" mailbox can send AND receive email while an "alias" can only forward email to a full mailbox or a third-party mailbox (e.g., gmail.com, yahoo.com, outlook.com).
For example, let's say you create the full mailbox "firstname.lastname@example.org" and the alias "email@example.com" which forwards mail to "firstname.lastname@example.org".
If someone emails you at email@example.com you can reply to them and they will see that it came from firstname.lastname@example.org.
If someone emails you at email@example.com it will go to your firstname.lastname@example.org mailbox. When you reply, they will see that it came from email@example.com, not firstname.lastname@example.org.
Aliases are meant to be used on websites to avoid having to change the email address all the time.
Consider this scenario
Jane starts working for you as a customer support representative. You update your website to let everyone know to contact Jane if they need any help.
Three months later, Jane decides to leave the company. But now, everyone is used to sending Jane their requests. Now you have to update the website, tell everyone to email Julie instead of Jane, and update any flyers or advertisements as well.
If you would have used an alias like sales@ or contact@ then you could easily send those requests to Julie once Jane leaves.
Why would I want sales@ to be a full mailbox instead of an alias?
You want to keep your 'real' mailbox private
If you're running the entire business on your own you might want to seem a bit bigger than you are by creating a real sales@ or support@ mailbox and responding to people from that address rather than your email@example.com address.
This is also handy if you want to limit the amount of mail you receive in your primary mail account.
You want to give multiple people access to an email account for sales or support
If you give the sales@ password to multiple staff members you can save money on mailboxes and they can answer your incoming requests without too much effort. They will each need to check the Sent messages to see if someone already responded to a request but this is an easy way to save on sales@ or support@ mailboxes.