I can't recall what was in the demo - I have some memory of the tutorial having quite a large night-time siege assault battle, so if there is something like that it would be a good test of what your computer can handle. Naval battles might also cause an issue for older computers.
In the past, I've found TW games can run on quite a wide variety of hardware by varying the graphics settings, but the quality of the experience is equally varied. ETW naval battles were agonisingly sluggish on my old computer and the land battles were ugly; on a new computer, they are both gorgeous. Try out different graphics settings and see how they feel.
My impression is that full versions of games often perform better than demos though.
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