There are no solutions, even when one of the fourth powers is replaced by a square. This was proved by Bennett, Ellenberg, Ng (see Int. J. Number Theory 6 (2010), 311-338).

P.S. I found the paper online here, I don't know how long the link lasts.

There are no primitive solutions, even when one of the fourth powers is replaced by a square. This was proved by Bennett, Ellenberg, Ng (see Int. J. Number Theory 6 (2010), 311-338).

Non-primitive solutions there are plenty, see Noam Elkies's comment for an example.

P.S. I found the paper online here, I don't know how long the link lasts.