The short answer is that the impact of firearms on warfare is due to much more than just improved armour penetration.

Some advantages of guns:

• better armour penetration

• greater effective range

• less training required for reasonable accuracy

• flatter trajectory, so less likely to be stopped by branches, foliage, etc. in forests

• more stopping power (important for large game)

• use is less dependent on physical strength - firearms can be used effectively by soldiers who are sick, weak from hunger, etc.

Some advantages of bows:

• higher rate of fire

• easier to reload on horseback

A major disadvantage of bows:

• archery requires long and dedicated training for effective use in battle

A difference, which can advantage either:

• cost of weapons and ammunition will differ, and ease of local manufacture of weapons and ammunition will differ.

From these various advantages and disadvantages, some benefits of guns over archery are clear: it can be easier to field larger armies, with less training required of musketeers compared to archers. On the battlefield, archers will shoot faster, but musketeers will often have a greater effective range.

The adoption of guns for hunting will affect their use in warfare - it will affect how many guns are available, and how many potential soldiers already know how to use them. Especially where large game (e.g., moose, bears) is hunted, the gun can be superior. The gun can also be superior in forests. Both of these factors are relevant to parts of Africa and North America. If warfare takes place in forests, the gun can be even more favoured.

Where (a) cavalry is widely used, and (b) potential cavalrymen can be expected to have pre-existing archery skills (due to use of the bow from childhood by hunters/herders), the bow can remain the preferred cavalry weapon. If cavalry sufficiently dominate warfare, this can slow the adoption of the gun. E.g., note the use of the bow as a major cavalry weapon by the Qing until the 19th century modernisation of armies. (It's possible to make use of the advantages of both gun and bow by carrying both - Tibetan cavalry and other Central Asian cavalry would sometimes carry both bow and musket.)

There are also some other points which should be stated:

Firstly, the adoption of the gun was often quite gradual, and only slowly changed the nature of warfare. The gun was added to the weapons in use, rather than replacing other weapons - it did not offer sufficient advantages to be a replacement.

Second, armour was extensively used in North America, until the growing use of firearms made it obsolete. Heavy rawhide armour (multiple layers, sometimes with hard layers of sand) and large shields were effective against archery, but not against guns.

Third, parts of Africa (e.g., the western Sahel, the Sudan) had strong cavalry traditions, including the use of armoured cavalry (with armoured horses).

Fourth, many groups in Africa and North America only widely adopted guns in the 19th century, and the guns that they adopted had the advantage of centuries of development over the earliest guns.

Fifth, if two neighbouring tribes/groups/peoples have been at war for a long time, and neither has been able to conquer the other, one reason is often that the two sides are relatively closely matched in military capability. If one side can gain an advantage, even if only a relatively minor advantage, it can be enough to upset the equilibrium, and allow the group with the newly-gained advantage to conquer the other.

Further reading:

North America: Roland Bohr, Gifts from the Thunder Beings: Indigenous Archery and European Firearms in the Northern Plains and Central Subarctic, 1670-1870, University of Nebraska Press, 2014.

Africa: Robert S. Smith, Warfare and Diplomacy in Pre-colonial West Africa, James Currey, 1989.