Proving Two Complexes are Not Quasi-Isomorphic In Richard Thomas' paper "Derived Categories for the Working Mathematician" he mentions (page 6) that the two complexes
$$
\begin{align*}
C^\bullet&= \mathbb{C}[x,y]^{\oplus 2}\xrightarrow{(x,y)}\mathbb{C}[x,y]\\
D^\bullet&= \mathbb{C}[x,y]\xrightarrow{0}\mathbb{C}
\end{align*}
$$
have the same cohomology but are not quasi-isomorphic. Proving that $C^\bullet$ and $D^\bullet$ have the same cohomology is straightforward, but I'm having trouble proving that they are not quasi-isomorphic. Any suggestions?
Thanks in advance!

As was pointed out by Andrea in the comments to the other answer, the other answer does not answer the question properly.
An argument that the two complexes cannot be quasi-isomorphic in the general sense is sketched on page 2 of:
https://www.math.upenn.edu/~tpantev/rtg09bc/secnotes/ainfnity.davidovich.pdf
