Are holomorphic an anti-holomorphic vectors or forms in any way fundamentally different? The question is stated in the title. Could one introduce all the complex geometry concepts just by using the "anti-" objects instead? Or is it just a $Z_2$ symmetry thing like it sometimes is with complex problems?

1. The identity map is holomorphic; it is not antiholomorphic.

2. If you pull a (p,q)-form by a holomorphic map, you get another (p,q)-form; but pulling it by an antiholomorphic map gives you a (q,p) form, an element of a different space. Etc. It's like comparing positive numbers to negative: as long as you stick to addition, there is a perfect symmetry, but once you start considering multiplication, positive numbers win.