I found in a book the following definition of $\succsim$-maximal element and $\succsim$-maximum, but I really don't see the difference between what is written in the book (I know the difference between those two concepts). Am I wrong?
Let $(X,\succsim )$ be a preordered set and $∅ = Y ⊆ X$. An element $x$ of $Y$ is said to be $\succsim$-maximal in $Y$ if there is no $y \in Y$ with $y \succ x$. If $x \succsim y $ for all $y \in Y$ , then $x$ is called the $\succsim$-maximum of $Y$.
On
A maximum element is necessarily comparable with all the other elements. It is possible to have two different maximal elements which cannot be compared with each other.
To illustrate take the set of integers greater than $1$, and $a \succsim b$ defined as $b=ra$ for some positive integer $r$.
Then $2$ and $3$ are both maximal, but neither is a maximum.
Add $1$ to the set, and $1$ becomes the unique maximum.
The difference in the negation:
Maximal element is one that there does not exist anyone strictly larger.
Maximum element is one that every element is smaller or equivalent to.
We have $\lnot\exists y\varphi(y,x)$ and $\forall y\varphi(x,y)$, where $\varphi(x,y)$ states $y\precsim x$.