Watching my dad playing scrabble, I noticed that his score was $243=3^5$ while the score of his opponent was $368=243+125=3^5+5^3=13^2+10^2+10^2-1^2$. So I came up with the diophantine equation in the title. More than knowing all the solutions thereof, I would like to know if interesting things can be obtained considering its symmetries, like $(a,b)\mapsto (b,a)$ and the possibility of representing $a^b+b^a$ by a quadratic form analogous to an interval in special relativity with differential elements replaced by positive integers.
So which non trivial piece of information about this equation can we extract from its shape?
Any integer can be written in the form $x^2 + y^2 + z^2 - t^2$, so there is no other non-trivial information.