I'm seeing a slight difference: "The students did the homework themselves." = without outside help. "The students did the homework by themselves." = also "without outside help", but "by themselves" can also mean "alone". The first example hints that the students could have worked together, however the second example hints that the students each worked separately. It's a really tiny issue, but generally you can use either one. :)

either one is correct

It depends on the idea that you are trying to convey. If you are trying to express the idea that the students did the homework independently without help, then "by themselves" is more accurate. The students did the homework themselves...is another form of -The students themselves did the homework. The pronoun "themselves" is used in this way for emphasis. -The students themselves (those very students) did the homework. This does not necessarily mean that the students did it "by themselves"; it means that the very students that you are talking about did the homework. It is possible they were helped, but it was those same students who accomplished their homework.

all by themselves

"by themselves" i more often used..