Is it really that simple? Does one have to use different statements in the <head> section for a php file? Are there any other differences I need to know about?
Yes it's really that simple, no, you don't use any different statements, and no, there aren't any other differences...

Does php recognize all html commands?
Here's where you are kinda missing it. PHP is a language that runs on the server. All of the commands are interpreted and executed there. If the php script says to do so, the PHP "engine" returns stuff to the web server that is in html, which is then sent to the users computer. Their web browser then interprets the html to decide how to lay out the page to be viewed. PHP is never sent to the user, while HTML is.
For instance, Anole's Weather Graphics script is several pages of PHP, and requires an additional text file of data, but all it returns to the user is a image, and that's all they see, even if they try to view the source code.
I made that little distinction there about "If the php script says to do so..." because it's entirely possible to write PHP that can be called by a browser, but doesn't return anything back. I have some that do that to calculate some daily weather indexes and then write a data file back to the server. While I used a web browser to test the script, in normal use it's called by Linux's cron scheduler just after midnight.
As you have figured out, you can rename an HTML file to .php and it still works. The .php is needed to tell the server what to do with the file when someone calls it.
Chris