Warre's is made by the same people that make Graham's and Dow's. While I have no experience with Warre's, I do like the ports under the "Graham's" name.
Tawny ports are those ports that are aged in barrels as compared to bottle aged ports. One thing regarding the age of tawny ports. The 10 year that you see is not the actual age, rather it is average age.
I don't know the differences in the black vs red cork however, here is Warre's website: Link
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