The only occasions on which a bow is defined solely as a weapon of war are the English longbow, some heavy bamboo or metal bows from India and the asymmetric bows of Japan.
The english yew bow was practised by townsfolk, merchant and commoners alike...not out of choice but by royal edict. It is a powerful bow, long ranged and throwing a heavy arrow, and even though it can be very accurate in the hands of an expert, it is, in reality, medieval artillery, used to "carpet bomb" an area of the battlefield. I challenge you to take such an overpowered, extremely large bow into a wild woodland to hunt roe dear. A smaller flat, self or composite bow is better in such situations.
As well as being powerful the medieval longbow is extremely simple in design when compared to say the holmegaard design that predates it by a couple of milenia.
The war bows of India are a rarity, perhaps due to their lack of durability (bamboo, though more durable than wooden variants in the west is still damaged by excess moisture) in a country that has a LOT of rain in season, or the steel versions which must have cost a fortune to make in their day. I imagine Indian history would be very different if such bows had been mass produced to counter the assault of mounted warriors that plagued the Indian sub-continent.
The Japanese bows of an asymmetric design are purely weapons of war, of a composite design, mostly of bamboo. To be used on foot or mounted.
On the subject of arrowheads, well this topic can, and has filled whole books. And whilst Rincewind42 has the right idea about differing function in arrowheads his observations are incorrect.
As for the mention of Mongolian archers, the concept of the bow to these people, descended from countless tribes who all venerated horses and bows above all else, is completely different to any of the other examples here. The Mongolians and their ancestors practically worshipped the bow (alongside the horse) as a weapon, a tool, for sport, as a symbolic representation of ones manhood and competence.
So @DVK the OP, no, a bow is a bow until the armies of the middle ages, feudal Japan or Gupta empire of India.(and some middle eastern similar)
Likewise an arrow is an arrow, balanced to the the weapon that casts it (search the archers paradox), with differing a head depending on the task at hand, materials/finances available or current trend. A bullet point bodkin for targets, a long tetrahedral or similar to pierce chain or even plate, or a broadhead to slash the arteries and vitals of prey. Of course there are many many more.
written by a historical reenactor, field and target archer and hunter.
PS. I dont consider crossbows to be the same class of weapons as bows. Crossbows DO show more polarisation depending on their intended use. Lever loaded hunting crossbows vs crank loaded and heavy stirrup loaded for war.