"Wurde" only means "was" if the sentence is written in the passive voice. This would only be the case if "wurde" were followed by the past participle of a verb, e.g. "Es wurde geschlagen" (It was hit). If what follows is an adjective like "kalt", then "wurde" does not indicate the passive voice, so it's used instead with its other meaning, the past form (became) of "werden" (become).
First of all, the difference: Your first sentence simply describes a condition. The second sentence refers to a change: It was warm last week, but then it got cold. / The cake was warm but it got cold before she came home.
Second, I think that the "get cold" is actually grammatically incorrect (though heard too frequently..) - really it has to be "it became cold".
It's not grammatically incorrect. See definition 3 of the ODO: http://oxforddictionaries.com/definition/english/get
or definition 5 in Collins: http://www.collinsdictionary.com/dictionary/english/get
Neither of these dictionaries even considers it "informal". One of the examples in the ODO is actually quite good because, at least to my native ears, "It's becoming late", though not wrong, would sound strange.
Thanks for correcting that! I checked it again and you are right. Maybe I should delete that part from my comment so that it won't be misleading?
However, my dictionary clarifies that "get" and "become" are not interchangeable, so some expressions only go with either one. It actually mentions "it's getting late" and I would agree with you that "it's becoming late" sounds a bit strange also to my non-native ears. :)
Sorry. What a mess - that wasn't even yaqo's question... ;-)