Understanding NIPALS algorithm for PLS I've found a nice presentation describing PLS1 and PLS2 algorithms (pages 16-19). It's pretty clear but there is a thing confusing me.
For PLS1. Let's look at the algorithm. The first steps are


*

*$w = X'y$ (which maximizes $\operatorname{cov}(Xw,y)$)

*$w = w / ||w||$

*$t = Xw$ 

*$p = X't / t't$


On the one hand one can say $T=XW$ and $WW'=I$, but on the other hand $T=XP$ and $PP'=I$ (because we are searching for a decomposition $X=TP'$ where $PP'=I$, see page 14). 
So, my question is, aren't $P$ and $W$ the same matrix? And if so, why does the algorithm need to calculate $p$ as $X't / t't$? 
Why not do it this way:


*

*$p = X'y$ 

*$p = p / ||p||$

*$t = Xp$ 


UPDATE
After reading provided comments, answers, and links (thanks to @amoeba and @theGD), I get that a strict answer to my question is "No, they are not.". I almost understood why. Actually, I lost my hope to fully understand NIPALS algorthm principle. So, I decided to ask it in a differernt way: What is mathematicial task PLS NIPALS tries to solve?
For example, there is NIPALS for PCA as well. And I don't understand it fully too. But I know that it's just a computational method for solving a mathimaticial task (for one iteration):
$$
{\mathbf  {t}}^{T}{\mathbf  {t}} = {\mathbf  {w}}^{T}{\mathbf  {X}}^{T}{\mathbf  {Xw}} \rightarrow \max, \textrm{ given that } \Vert {\mathbf  {w}}\Vert =1
$$
So, what is analogous mathematical task for PLS?
 A: The 14th page is not about PLS, it is about PCA since it refers to decomposition of $X$ alone by carrying out singular value decomposition which is an efficient way to obtain eigenvalues($D$) and eigenvectors($V$) of $\operatorname{cov}(X)$). So in PCA case, the orthogonality of $P$ is correct.
In PLS, however, the only property of $X$ loadings is each vector in $P$ matrix has unit length ($||p_i||$ = 1) whereas $W$ is orthogonal. In fact, the addition of $W$ is to ensure the orthagonality of $X$ scores($T$) and it is one of the main differences between PLS NIPALS and PCA NIPALS. Thus, as said in comments, $P$ and $W$ are not the same matrices.
As a small note I prefer SIMPLS algorithm which is faster and it provides a single vector for regression and makes interpretation a lot easier unlike NIPALS with multiple vectors each corresponding to deflated $X$ which is, in my opinion, quite counter-intuitive and hard to interpret.
The properties of PLS factors obtained by NIPALS algorithm can be found in this article:
Geladi, Paul, and Bruce R. Kowalski. "Partial least-squares regression: a tutorial." Analytica chimica acta 185 (1986): 1-17.
