Yellow VS Black

Observe that if a black square has the number $n$ , then the square with the number $n+1$ is yellow.

In this way, we can pair up all of the black squares with a yellow square that has a larger value. More explicitly, the black $2n$ pairs up with the yellow $2n+1$ .

Hence, the yellow squares win.

Relevant wiki: Arithmetic Progression Sum

Notice that the sum of yellow squares is the sum of the terms in the arithmetic progression:

$1,3,5,\ldots,25$

There are $13$ yellow squares, therefore

Sum of yellow squares $=\dfrac{13}{2}(1+25) = 13(13) = 169$

Similarly, the sum of black squares is the sum of the terms in the arithmetic progression:

$2,4,6,\ldots,24$

There are $12$ black squares, therefore

Sum of black squares $= \dfrac{12}{2}(2+24) = 12(13) = 156$

Therefore, the $\boxed{\text{Yellow Squares}}$ have a greater sum