According to Donnellan, a speaker who uses the description 'the F' attributively means to state something about whoever or whatever is F. A speaker who uses the description 'the F' referentially uses it to pick out who or what he is talking about. By implication, then, the attributive use does not pick out who or what we are talking. Probably the most well-known claim of the paper is that we can use a description referentially even when nothing fits the description. Donnellan gives the example of 'the man with the Martini', which could be used to successfully pick out someone who (in fact) is drinking water, not Martini, and to say something (according to Donnellan) which is true. For example, we could truly say 'the man with the Martini has glasses' in the event that the man drinking water has glasses. I noted with interest that Donnellan attributes this idea to Leonard Linsky - we can say of a spinster that 'her husband is kind to her', referring to her lodger.
I don't understand the distinction between referential and attributive at all. In the sense that 'referential' means that the semantics of the referring term is dependent on the existence of an object referred to, I hold that no terms are referential. Meaning is independent of external reality. In the sense that 'referential' means picking out or signifying which individual we are talking about, all meaningful singular terms do this, as I argued here and particularly here. Even 'the King of France' - Donnellan's favourite example of an attributive use - tells us which king we are talking about. Not the King of England, nor of Ireland, nor of Spain, but the one who rules France. Of course, there is no such person as the King of France, but even as I say this, I have just told you which king there is no such king as.
So Donnellan's distinction is artificial. And certainly it cannot be invoked to explain why the following argument is valid in one sense, invalid in another.
The prime minister of the UK is a manVallicella means (I think) that the argument is valid if the description 'the prime minister of the UK' is used referentially, i.e. to refer to Cameron, invalid if not. By contrast, I say that the argument is valid when the description has the same sense in both premisses (meaning 'the UK Prime Minister in 2011') but invalid when it has different senses. If it means 'the UK Prime Minister in 2011' in the first premiss, but 'the UK Prime Minister from 1979-1990' in the second, then it is invalid, just as any argument is invalid when the middle term is used equivocally.
The prime minister of the UK used to be a woman
Some man used to be a woman
*”Reference and Definite Descriptions”, Philosophical Review, 75 (1966), pp. 281-304.