No - it's a difference between expressing shared or unshared information.
The definite article "the" introduces information which is shared between the writer/reader or speaker/listener (have a look at
this thread for more on this). The indefinite article "a/an" on the other hand, introduce information which the writer/speaker assumes will be new to the reader/listener.
Here, Orwell assumes you know who Churchill is - his role as Prime Minister is assumed to be shared, and therefore "the" is used. Compare it with :
Andrew Law, a former Prime Minister of Britain, once said that...
This time, I'm not assuming that you know who Law is - in fact I'm assuming that the information will be completely new to you. And so I use "a".
Another example would be :
I spent my holiday in Dax, a town near the south west coast of France. (I presume you've never heard of it)
I spent my holidays in Washington, the city on the east coast that is, not the state on the west coast. (I presume you already know of both).
To say
Winston Churchill, a former Prime Minister.... would therefore sound as if you imagined the reader had never heard of him and needs to be given information on who he was. Which in this case would probably sound as if you were insulting the reader's general knowledge.