To return to your question, the technology was exactly the same, but the construction and details could be different. Military firearms were "mass produced" (as much as that term can be applied in JA's time period), and were intended to survive fairly rugged treatment. This meant that they were fairly heavy. Cost was also an important factor. If you look at the various models of the "Brown Bess" during the Napoleonic period, you will notice they were a little less ornate, and reduced the amount of brass work as war dragged on.
Hunting arms were not made in such large numbers. They did not need to be as rugged as the miliary weapons. They aren't expected to be out in the same elements for a long time period (s soldiers would be). Nor were they made in quite the same numbers. Far fewer people had cause to own a firearm in Great Britain than they would in the wilds of the colonies. Cost was also not so much a factor. Gentlemen of means would buy from the best gunsmiths, possibly having a rifle or shotgun tailormade to his size, tastes and other requirements. Some of these could be as much a work of art as a means of slaughtering the wildlife.
On a side note, it should be pointed out that longarms were either "rifled" or "smoothbore." Rifles have groves in the barrel which give the musket ball (so called because they are spherical) a spin when it is fired. This helped stabilize the ball in flight helping to improve accuracy. Rifles would be used for hunting larger game (deer), or amongst sharpshooting units in the army (the 95th Rifles as portrayed in the Sharpe's series).
Smoothbores do not have this rifling, and are more like a modern shotgun. One could put a quantity of shot down the barrel for hunting birds or other small game, or a musket ball. These muskets, used by the bulk of the British Army, were far less accurate. It is not likely that a hunter would use one for large game if he had a rifle. It is possible to improve the accuracy by using a tight fitting patch to give the ball a snug fit. However, this method slows down the loading time. This is also a problem with rifles, especially as with repeated firing the inside of the barrel becomes "fouled" with the residue from the black powder. If the fit of the ball is looser, it can be loaded far more quickly, 3 or 4 times a minute instead of one or two. This is part of the reason why the armies of this age fought in such tight formations. The rationale being to put as much lead down the field as quickly as possible.
It appears I've rambled on a bit and should stop before I get too carried away. I had intended to be quite brief - you shouldn't have got me started. ;-) To sum up for a short answer: civilian firearms were: lighter, better made, and more expensive than their military counter parts.
I remain,
Jason E.