The difference between the Très Vieux Marc de Champagne and Très Vieille Fine de Marne is that the Fine de Marne is made of distilled wine and champagne and the Marc de Champagne is a traditional Eau de Vie (brandy) that is produced by distilling the grape skins, seeds and stalks, which are left from the pressing process in the first stages of Champagne production. Marc is much stronger in taste while the Fine is finer in taste (closer to a Cognac) and is often used for making chocolate truffles.
Marc de Champagne is aged in oak casks and reduced to an alcoholic strength of 40%. In France, it is usually drunk at room temperature, in a round wine glass so that all the subtle and warm flavours can be appreciated. In other countries, such as Spain or Italy, it is often drunk chilled or on the rocks. It is also used as a base in a long drink. You can also use this Eau de Vie in cooking, to flavour sauces, ice creams, sorbets or even to flambé meat.
Très Vieux Marc de Champagne is enjoyed during a French meal as a traditional "trou champenois" (digestion aid). It is also ideal with a sorbet, ice cream or coffee.
Très Vieille Fine de Marne is quite a rarity nowadays. Fine is the French word meaning "fine", as in "high quality" and is a term for some high quality French brandy (generally AOC), not including Cognac and Armagnac. Varieties include: Fine de Bordeaux, Fine de Bourgogne, and Fine de la Marne. People used to refer to having a couple of fines after their coffee but the term, though once common, is now dying out.