shel, being that a mushroom-barley soup I would recommend to substitute for a medium sherry, tawny port is sweeter than medium sherry, but that might be enough sweetness anyway. Tawny port is more amber, for taste and in a mushroom soup I would go first to a sherry, then to what you think of as regular port which is probably LBV (late bottled vintage) or ruby, those would be further down the list to me as a sub for tawny in mushroom soup, they are more of a claret red. (so is vintage port or single quinta vintage but I'd drink those, or if I was going to cook it would have to be something pretty fine for me to want to use it in that way).
I find the tawnies too sweet for my taste. KYH hit on the basic differences, you can think of tawny as your "regular" port plus more wood and oxidation to brown-ish from red.