Game and Coarse Fishing
The first use of these two terms was made more recently than many people think, during the late nineteenth century – prior to that, it was all just fishing. The origin of the word "coarse" is almost certainly related to the value of the fish concerned as food items; few British fresh water fish other than salmonids make good eating without a great deal of preparation and although species like bream were common table items in the Middle Ages their flesh is frequently described by older writers as being coarse. The middle of the nineteenth century heightened this awareness as a switch occurred away from the centuries’ old menu of pond-reared freshwater fish, such as carp, tench and pike, towards the tastier and increasingly available sources of fresh sea fish and iced salmon.
Needless to say, if you go more deeply into the matter, the explanation turns out to be a bit more complicated than that. There were several reasons for the schism: on the one hand, by 1825, the increasing pace of development meant that fly fishing and ‘bottom fishing’ (for want of another word, other than coarse) gear was quite different, which underlined the difference between the two methods and meant that anglers were increasingly likely to fish with one or the other method. The term "all-rounder" first came into use during this period and it underlines the awe in which fishermen who were good at both methods were held, because an increasingly large amount of knowledge and skill were necessary.
Another catalyst was almost certainly the way increasing wealth and the ability to travel, thanks to the railway, meant that the emerging middle class could take itself to Scotland and go fishing for salmon, using flies like the Britannia, shown here (tied by Alberto Calzolari). This group of anglers was socially quite distinct from any that had gone before it and in that many of them identified themselves as "field sportsmen" and fished, hunted or shot driven pheasant as the seasons changed – these were the people who began to use the words game and coarse to describe fishing and some of our greatest angling writers, like Francis Francis, Cholmondeley-Pennell and William Senior were among them. The word "game" came naturally to them, as a generic description of creatures that could be hunted and the word "coarse" gained currency by default; a term was needed to describe all the species which weren’t "game" fish.
Although there is an argument for a different term nowadays, mainly because we no longer use the word coarse to describe food, at the time of the split few objections seem to have been raised; Greville Fennell, the famous roach angler, was using the term by the early 1870s and angling club members routinely described themselves as coarse fishermen by the 1880s.
At the risk of repeating myself, mainly for the benefit of American readers, who don't have to live with this artificial division: deciding whether a fisherman is a "game" or a "coarse" angler is determined by his or her quarry, with salmon and trout anglers being classed as game fishermen and hunters of all the other species as coarse anglers. No-one has ever really decided which side of the line grayling lie, so some regard them as game and others as coarse fish. At least in theory, the method doesn't matter, so you could float fish for salmon and still call yourself a game angler (although in practice, you will get called all kinds of other things) and equally, if you went fly fishing for roach, you would be a coarse angler, because you were after a coarse fish. But no-one has ever written any of this down and no law relates to it, which is why the angling license refers only to migratory and non-migratory fish - when all is said and done it is the riparian owners or clubs who decide which methods are allowed to catch which species in their waters. For what it is worth, there is no reason why you shouldn't float fish for salmon - I tried it when I was a kid and it works, but it raised a few eyebrows. Legering for salmon with worms is also tremendously effective, but the method is generally described as poaching!
So the terms might not be ideal and neither are they particularly descriptive, but they have stuck and they are one of the few things that the general public actually understands about angling.