The difference is that the pudding is eaten hot, usually brought to the table flaming as you traditionally cover in cognac or rum or whisky and light it!
Then you have it with cream, custard, ice cream or brandy butter. The tradition also is that you hide some coins in the pudding (either in one of the slices or when you make it as a raw mix) and the lucky people get the coins.
It's really the dessert that follows the Christmas lunch.,
A Christmas cake is part of "high tea". So in the afternoon early evening after your turkey sandwiches, sausage rolls etc you have a slice of cake. It's cold and usually covered in a layer of marzipan and icing and some festive decoration. Or can be fruit and nut topped.
Because you "feed"'the cake regularly with alcohol up to the day you ice it (I usually ice it a few days before Crimbo) it is saturated with alcohol and won't go off if kept air tight.
Usually it's sliced on the night of Christmas but often people give a slice to Father Christmas on the 24th. I usually keep mine in a cake stand and then offer for the rest of the season.
You can actually keep a cake for a whole year!!! I made two last year, fed them well and have one left for this year!
But to be honest you'd never offer a slice of cake after lunch traditionally. Only pudding.
Me....I'll take a mince pie any time.
I'm eating them for breakfast now