What is there difference between these two expressions, or are they synonmyous?
There are, in fact, two different words written 别人.
The first is pronounced "biérén" and is a noun. It means "other person (other people)" (另外的人). An example of its use is a sentence like
屋子里只有我和他，没有别人。 Only he and I are in the room, no one else.
The second is pronounced "biéren" (note the neutral tone on the second syllable) and is a personal pronoun. It is used to refer to people other than oneself or the person in question. Dan provides a good example (though I would have used 扔 instead of 丢 -- a native speaker will have to weigh in here); in "是别人丢的", 别人 is a pronoun.
As far as I know, 别的人 can be used in place of 别人 in both cases. Nevertheless, I think 别人 is more common in general, and for some reason I tend to think that 别的人 is more naturally a noun than it is a pronoun.
I'm not a native speaker, but I think they can be used the same in most cases but in some cases are slightly different.
Did Tom's brother throw the ball?
Is Tom's brother the same guy who threw the ball?
不，Tom 的哥哥是别的人 (I don't think 别人 would work as well here)