The logical assumptions built into the periodic table as humans use it are as follows:
1) Sorting a list of the base elements of matter makes most sense if done by atomic number, with groupings by chemical properties affecting the structure of the table. I can think of other ways to organize things. For example, maybe the aliens structure things by base state (solid, liquid, or gas) at a specific temperature, and include simple molecules like H2O in the categorization? Or...maybe they prefer to focus on isotopes? They may have a strong preference for listing elements in the order in which they were first created in pure form. Maybe the civilization is so focused on metallurgy that their equivalent of the mohs scale is the most important aspect to them?
2) The aliens want to categorize things as a way of drawing conclusions. That's how human psychology works, to the point where its difficult to conceptualize a different way of thinking. What if the aliens function primarily through kinesthetic patterns? They know about all of the common elements, but their version of a periodic table would be a description of the process for creating pure versions of each element.
3) The aliens use a visual writing style in a range visible to humans. I could easily see key parts of the table shifting into infrared, or the whole thing being in alien braille, or based on chemical signatures impregnated on slates.
4) The aliens would use a grid-type layout with square blocks. Humans seem to prefer angular geometry, but what's to say an alien wouldn't prefer a spiral out of a central point?
So, yeah, there are a lot of things you can invent as to why a periodic table on an alien planet wouldn't serve as a very good Rosetta stone, as it might not even be recognizable as such by Earth explorers.
