The meaning is the same. If we compare できるだけ and できる限り, できる限り (literally “to the limit one can”) sounds stronger than できるだけ (literally “as much as one can”), so the nuance is slightly different, but not much.
While onteria_’s answer and the answers on Goo to which it links have some points, I think that they are describing the difference between the two phrases bigger than it actually is.
I wrote above “If we compare できるだけ and できる限り, できる限り sounds stronger than できるだけ.” But when we hear one of the phrases, it is rare to compare it to the other phrase which was not used.
I do not buy the argument that できるだけ implies the possibility of cutting corners. Sure, it is true that “depending on the situation, できるだけ sometimes gives the impression of cutting corners,” of course, but that is saying nothing. The same applies to できる限り, and it does not explain any difference between the two phrases.