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.