As they are defined, they seem to be the same thing to me, other than that a pre-lattice involves finite subsets... Is there another important difference?
These are the definitions in the text:
We say that a partially ordered set $( S, \leq)$ has the largest-lower- bound property if $\inf E$ exists for every subset $E \subseteq S$ which is nonempty and bounded below.
Dually, we say that $S$ has the least-upper-bound property if $\sup E$ exists for subset $E \subseteq S$ which is nonempty and bounded above.
Partially ordered sets with the largest-lower-bound property are said to be inf-complete, and those with the least-upper-bound property are said to be sup-complete.
A partially ordered set $( S, \leq)$ is called a pre-lattice if every nonempty finite subset $E \subseteq S$ has supremum and infimum.