Both adjectives mean the same. 'Britannic' is dated and now used only in titles like 'Her Britannic Majesty'. It sounds much grander than 'British'.

I think there is no difference between them. Bristish, Britain, Britannia or Britannic. I remember Britannic is also name of a lost Ship in Sea.

There is no difference.

@alex, thank you!!!

I think there was an ocean liner named the "Britannic". As to your question, they mean the same things, but the term "British" is used almost every time while if you use the adjective 'britannic' you might get an amused reaction.