Brit is definitely a commonly used word for describing people from Britain. While it may turn up in the occasional tabloid headline, it is mostly used by 'non-brits'.
I am English and personally I don't like to be considered a 'Brit' because it has several negative connotations, one example being 'Brits Abroad' which refers to the British hooligan holidayers who travel to popular beach destinations and get drunk and disorderly.
For this reason, I prefer English. I would also suggest that citizens of the other British countries would also prefer to be referred to by their own nationality (Irish, Scottish, Welsh) due to various conflicts in the past (generally with England).
As for formality, it is definitely informal, having only come into use in the last century.
Hope this helps.
N.B. As an adjective, 'British' is more applicable since 'Brit' is the noun. The same can be done with 'Scottish' and 'Scot', with 'Scot' being a noun.