Two things. One, I think the normal rules say somewhere that an unarmed strike is generally a punch, but I dunno if that's the case (can't remember), whereas monks have the specific advantage of being able to execute unarmed attacks with any part of their body.
Two, while a non-monk can probably do what you describe, he or she would be suffering the penalties for two-weapon fighting, and the unarmed strike would be an off-hand attack (so only half Strength bonus to damage, and whatnot). Whereas a monk can strike with a quarterstaff two-handed, then headbutt the enemy or kick them or whatever, and he gets his full Strength bonus to damage with the unarmed attack, AND the monk doesn't suffer TWF penalties for doing so (because he can use any body part for unarmed strikes as though it were a normal primary hand attack).
I'm still fuzzy on whether or not a monk can combine TWF with unarmed strikes (frex, to make 3 attacks at 1st-level, two using flurry of blows and one from TWF, though suffering a nasty penalty on the attack rolls), and whether or not he or she specifically has to use monkish weapons to do so (rather than just TWFing with unarmed strikes alone). The wording involved was confusing somehow, as I recall, but I had enough trouble hitting things with my monk, Argus, in For More Than Glory. So I hardly used anything but unarmed strikes with flurry of blows, or just my quarterstaff with TWF and nothing more (this was before I learned that 3.5 monks can use a quarterstaff with flurry of blows no problem).