For example, in $\Bbb R^n$, both
$$\frac{\langle v,w\rangle}{\|w\|^2}w \qquad \text{and} \qquad \frac{\langle w,v\rangle}{\|w\|^2}w$$
are projections of $v$ on $w$, but in $\Bbb C^n$ only the first is right. What makes this difference?
I've found an answer here, but I'm still don't understand why the first is complex linear, whereas the second is complex-skew-linear would make the difference. And help, thanks!
Actually it depends on a convention about linearity of the Hermitian form. Usually, mathematicians use the 'first entry convention', i.e., the first entry is linear, while the second entry is antilinear:
$$\langle cv,w\rangle=c\langle v,w\rangle,\qquad\langle v,cw\rangle=\bar c\langle v,w\rangle$$
Physicists, on their turn, often use the 'second entry convention', i.e.,
$$\langle v,cw\rangle=c\langle v,w\rangle,\qquad\langle cv,w\rangle=\bar c\langle v,w\rangle.$$
So, in the mathematicians convention, your first formula is the right one for the orthogonal projection onto $\Bbb Cw$, because the orthogonal projection is normally assumed to be linear and not antilinear, while in the physicists convention it's the second one. I have no idea of the historical reasons for this discrepancy, though.
Anyway, it's probably possible to consider a somewhat unorthodox 'antilinear orthogonal projection' and see what happens, but I believe that kind of trouble isn't what you are looking for right now! =]