Give each input a name attribute. Only the clicked input's name attribute will be sent to the server.
<input type="submit" name="publish" value="Publish">
<input type="submit" name="save" value="Save">
And then
<?php
if (isset($_POST['publish'])) {
# Publish-button was clicked
}
elseif (isset($_POST['save'])) {
# Save-button was clicked
}
?>