Lattice and order theory are often mentioned together. So I wonder how lattice theory is "intended" to differ from order theory.
- Lattice theory became important in the context of universal algebra. The characterization via algebraic operations instead of the order relation might hence be the important point.
- The generalization of intersection and union in set theory, of "and" and "or" in logic (and maybe of the analogous operations for "quotient structures") might also be the motivation for investigating lattices. But even so these are contexts where the non-trivial results of lattice theory matter, I don't have the impression that introductions to lattice theory care much about these "applications".
The difference between these two possibilities can be seen by looking at more concrete question regarding the scope of lattice theory: Is semi-lattice theory part of lattice theory? Is the "logic of partitions" part of lattice theory? Is formal concept analysis part of lattice theory?
Both semi-lattice theory and formal concept analysis are definitively within the scope of "algebraic order theory", but the logic of partitions is not. If on the other hand the generalization of logical operations were the central theme of lattice theory, then the logic of partitions would be part of it, but semi-lattice theory would not, and formal concept analysis would be a border case.
(I hope it became clear enough in which sense I'm "confused" about the scope of lattice theory.)