Perhaps all afternoon refers to today and all the afternoon refers to an afternoon in the past. It's been raining all afternoon. On Tuesday, I spent all the afternoon doing my tax return.
This is only a suggestion. Does Longman's say anything about this, Zaffy?
A pity. I take Barque's point, although I wouldn't go all the way with him. As I say, I'd reserve all the afternoon to refer to a day which isn't today.
I'd say It rained all afternoon on Christmas Day, but I don't think I'd regard it as a mistake if someone said ...all the afternoon...
Perhaps it's like on Tuesday/on the Tuesday. On Tuesday means last Tuesday or next Tuesday. (You can tell which one by the tense of the verb.) On the Tuesday means some other Tuesday. We went to London for Christmas and we went to the British Museum on the Tuesday.
We went to London for Christmas and we went to the British Museum on Tuesday means we went to London for Christmas (which might have been six months ago) and we went to the British museum on Tuesday this week (or last week if you're speaking on a Sunday or a Monday).