There seems to be some confusion about what is being discussed here.
Half of the answers are using the conventional definition of "Formal Writing", which involves the solemn and decorous application of language to convey the writer's authority as well as the meaning of their message. Formal writing, in this context, is an alternative to the casual or colloquial language of standard prose. Standard prose, by comparison, excels in blatantly expressing the writer's emotions, in addition to their message.
Each method of writing serves a specific purpose. Both should be familiar tools in the hands of any seasoned author.
The other half of the answers seem to be addressing, I believe correctly, a new definition for this established term. Here, "Formal Writing" implies any use of the language, which obeys the communally held standards of grammar, punctuation and spelling.
The alternative to this definition ignores those standards in favor of minimalistic phonetic renderings. I fear that the contractions refered to in this question are not the apostrophy-ridden abbreviations of standard colloquial language, but are instead, the attrocious missing-letter mis-spellings championed by Twitter and AIM. "R u w/me?"
In my opinion, this question explores the wisdom of maintaining minimal expectations for the competency of professional writers, in a world where the majority of readers no longer know how to recognise or participate in those minimums. This is a valid question and in an attempt to answer it here, by example, I have endeavored to write in the highest, formal english which my limited vocabulary allows...
Have I conveyed anything beyond the specific words of my message?
Have I established my authority on the subject of the English language and its evolution?
Have I, subtly and politely, expressed my anger towards where our culture and our language are heading?
Twitter-ish, to coin what I believe is a new term, is an attempt to portray the most basic meaning of a message in the minimum number of characters, digits and symbols.
English, by comparison, is the art of communicating that basic message and much, much more.