Polyurethane is essentially a coating that sits on that surface, but really has little penetration.
Thompsons is a penetrating stain, which wicks into the wood fibers.
Depending on your desired results, each has pros and cons.
If you have existing staining, then the wood must be prepared before you do anything.
A weak bleach solution and a stiff bristle scrub brush should do a fair job of it, assuming you're not super concerned about the quality of finish.
Then let it dry well overnight, and check it. If it feels cold, it's still wet, and you can't use anything that seals on it until it's totally dry.
You can, however, use a penetrating stain, as it leaves the wood pores open for ventilation.
Let that dry two days, so the voc's vaporize completely, and you're ready for poly.
I like an oil based type, my personal favorite being Varathane.
It gives a great bite into the wood, penetrates if you cut it 15%, and leaves a very hard surface layer.
It takes forever to dry. I like to give it 3 days to fully cure before I do anything with it.
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