Is there any difference between these two?
She was a good student, finishing in half the time the others took to finish.
She was a good student, finishing in half the time the others to finish took.
There is a difference, but not much; both are technically grammatical and a little awkward, but the second one is more awkward and a little less clear.
A more natural alternative than either of those would be
She was a good student, finishing in half the time the others took.
The extra words add next to nothing in this case, since they are already implied by context. In more complex sentences, they might be necessary, but the addition of them would have to be weighed against just making the sentence simpler by rephrasing. (English removes words in cases like this — which is called ellipsis — more often and in a few other cases than some other languages.)