Archive usually keeps the mail in Archive Folder, for future reference. Delete commonly puts the mail in a Trash folder, that is periodically cleaned, and can also be manually deleted forever on demand. You can access both of those folders from the Folders pane.
To reach either one, tap your account avatar at the top-left to open the folder list, then choose Trash or Archive. On a Gmail account the archive folder is All Mail. See Navigating between folders.
