What are some uses of the Thue-Morse sequence in computer science?
I don't know if this counts as an application but at least it shows up. When using a polynomial rolling hash, it's tempting to do it modulo $2^{32}$ or $2^{64}$ (depending on the word size of the computer) since on most modern architectures addition and multiplication of integers just handle overflows this way, saving time. This method will fail often on Thue-Morse-like strings (like ABBABAAB...), as explained here. I remember not understanding the factorization of $T$ a few years ago, the key is the recurrence relation $t_{2n} = t_n$, $t_{2n+1} = 1-t_n$ 

Alternative explanation of the factorization: from the "append the negated sequence" definition one can directly see $\Pi_n (1-x^{2^n})$ , since each each term does exactly that.