difference between maximal element and greatest element

You are maximal when there is nobody above you.

You are greatest when you are above everyone else.

Examples:

• If nobody has eaten you, it doesn't follow that you have eaten everyone else.

• If nobody is standing on your head, it doesn't follow that you are standing on everyone else's head.

• If you live on the top floor of your apartment building, it doesn't follow that you live above everyone else in the city.

So maximal elements need not be greatest.

(Incidentally, although a greatest element is always a unique maximal element, it is a fun exercise to come up with a partial order having a unique maximal element, which is not greatest.)

The crucial issue is the difference between a partial order and a linear order.

A partial order is a set where sometimes you can say "this thing is bigger than that", but some things are just incomparable. In a linear order, you can always say "this thing is bigger than that."

For example, if you're working with the natural numbers (except $0$), $\{1,2,3,...\}$, and you say "I'm going to declare that $n \leq m$ iff $n|m$. Then, reflexivity is clear ($n|n$), antisymmetry is clear (if $n|m$ and $m|n$, then $n = m$. And transitivity is also easy to check ($n|m$ and $m|p$ implies $n|p$). So, this really is a partial order.

Now, in this notion of order, is $2\leq 3$? No, since $2$ doesn't divide evenly into 3. Well, is $3\leq 2$? Again, no, since $3$ doesn't divide evenly into $2$. The conclusion is that we simply have no way of comparing the sizes of $2$ and $3$ in this order.

Now, to answer your actual question, as Stefan has noted, in a linear order (where any two elements are comparable) the two notions coincide.

In a partial order, we can see the difference. An element is maximal if it's bigger than everything it can be compared to, but we're not claiming it can be compared to everything.
An element is greatest if it's bigger than everything it can be compared to, and we can compare it with everything.

Suppose $\langle A,R\rangle$ is a partially ordered set (i.e. $A$ is non-empty, $R$ is a partial order relation on $A$).

An element $a\in A$ is called maximal if $\forall b\in A(aRb\rightarrow b=a)$. That is, there is no one "above" $a$ (except perhaps $a$ itself).

An element $a\in A$ is called maximum or greatest if $\forall b\in A(bRa\lor b=a)$, that $a$ stands "above" everyone in $A$ in the relation $R$.

Note that both these definitions hold whether or not you require $R$ to be reflexive.

From this follows that a greatest element is by definition maximal, but not vice versa.

Consider the following case $A=\{a,b\}$ and $R$ is defined to be the identity relation, then both $a$ and $b$ are maximal, but neither is the greatest.

Another strong example with a single maximal element, but no greatest element is this: consider the set $\{a\}\cup\mathbb Z$, and the relation $R$ defined to be $<$ for integers and $aRa$ otherwise (i.e. $a$ does not stand with the integers in this relation). In this case $a$ is trivially a maximal element, no one is "above" it, however there is no maximum element, since no one stands over both $a$ and all the integers.