TITLE: Waring rank vs tensor rank of symmetric tensors?
QUESTION [6 upvotes]: Suppose we work in an algebraically closed field. Then, do the Waring rank (symmetric tensor rank) and tensor rank of a symmetric tensor coincide in general? Recall that tensor rank is rank with respect to the Segre variety and Waring rank is rank with respect to the Veronese variety.

REPLY [10 votes]: A counterexample in $\mathbb{C}$ is given in A counterexample to Comon's conjecture:

We present an example of a symmetric tensor of size 800×800×800 which
  can be written a sum of 903 simple tensors with complex entries but
  not as a sum of 903 symmetric simple tensors.

It's a very recent result by Yaroslav Shitov, not yet published but it has survived some scrutiny by the tensor community.