So my question is: is there any difference between "Friend Computer" and "The Computer"? Or are they both referring to exactly the same thing, and both can be used completely interchangeably?
They are interchangeable except in terms of nuance. This is like asking "Is Mao (Tse-tung) really called Mao or Comrade Mao?" Well, both are frequent appellations, but you generally use Comrade Mao when you're on board with the whole Communist program. Similarly, The Computer is The Computer. Alpha Complex propaganda insists "The Computer Is Your Friend." So when talking to The Computer, or where The Computer might hear, it's appropriate to say Friend Computer. If you are in your treasonous secret society anti-Computer cell, you probably would just say "The Computer" or some other more negative appellation.
Alpha Complex is governed by an entity you can refer to as "Computer", "The Computer", "Friend Computer", "The Friend Computer" or [redacted for security reasons].
Different NPCs and organizations will have different preferences about which term to use. In the spirit of Paranoia some might have very strong (but still frequently changing) preferences and no qualms to declare anyone a traitor who uses the "wrong" term.
"The Computer" is what you'd probably say as an out-of-setting 3rd party narrator. "Friend Computer" is more likely the address by an in-setting character.
By comparison, the Wikipedia article on the current Queen Elizabeth usually refers to her as "Elizabeth", but if you met her in person then you're expected to address her as "Your Majesty".
If you don't consider the computer your friend, and visibly/audibly encourage others to do so, you're obviously a traitor. I'll need your ID code, so I can report your traitorous thoughts to Friend Computer. If you cooperate, and provide a list of all your friends' ID codes as well, it's possible that Friend Computer may be lenient...