I saw Loren's original post in this discussion; it looked fine to me, so I'm not sure about the misread.
You're not doing the subtraction properly in the first factor. You subtracted a negative z^2 term, but wrote it as -z^2.
Here is a symbolic example of the relevant rule when subtracting terms within parentheses:
A - (B - C) = A - B + C
The C ends up being positive, not negative. Subtracting a negative C is the same as adding a positive C.
You can think of that first minus sign as a -1 multiplying the contents of the parentheses, like this:
A + (-1)(B - C)
Now use the distributive property to get rid of the parentheses:
A + (-1)(B) + (-1)(-C)
A - B + C
Same result either way.
In the second factor, the -z^2 is not being subtracted, so I'm not sure why you changed it to +z^2.
I'm going back to your first result:
[(x^2+ y^2) - (y^2 -z^2)] [(x^2 + y^2) + (y^2 - z^2)]
Remove the inner grouping symbols.
(x^2 + y^2 - [y^2 - z^2])(x^2 + y^2 + y^2 - z^2)
Subtracting -z^2 is the same as adding +z^2.
(x^2 + y^2 - y^2 + z^2)(x^2 + y^2 + y^2 - z^2)
Let us know if you still do not understand why it's +z^2 in the first factor and -z^2 in the second factor.