It all depends on the general and the state.
The thing is that, generally speaking, a state with access to more advanced weapons would ALSO have access to more advanced warfare theory and thus that turns into more success in every aspect.
Let me give you an example. In the Fallout universe a very well educated man was captured by a tribe of backwards people. However he was able to convince them to follow him and in no time he was their chief. He then used his knowledge to raid other tribes and expand his domain and finally he created what is called Caesars legion. Read the wiki for more.
So what he did was not provide them with cutting edge tech as much as small improvements in the overall society and the theory behind warfare.
So going back to your example. Would a state just possessing far superior weapons be able to translate that into the battlefield?
Depends on the general I say.
The historical examples of an army with superior number or tech that failed to wrestle an advantage from an inferior force, on paper, are far too many to start mentioning.
You see warfare is all about maximizing your advantage. You know what they say: if you ever send your soldiers into a fair fight, you are a bad general.
So you could actually incorporate that into the story. How?
Well. Think of this as an admiral Yi or Sun Zu or a similar general who is really good in warfare and planning. Hannibal is that different as well.
So your general X is commanding a force of lesser technology and ever lesser numbers. Yet using the principles of warfare correctly and brilliant maneuvers and loads of deception...etc he is capable of beating the crap out of empire B with all their endless legions and superior tech.
So you have a scene where the "evil" generals are all setting around looking "evil" and bragging about their superior numbers. They even get bored of him and command a force that is 4:1 his army size. Now the rest is up to you I say.
And overzealous wing commander charges the enemy and ruins the battle plan, the cavalry excel and they win the day...etc.
Now that leads us into the obvious question. If 2 generals of comparable strength met, what will happen?
Well. History would also tell us that the experience of the soldiers and officers as much as anything is a far better weapon that mere weapons.
Moral and maneuvers and all that are the actual critical elements in war.
So even small differences in those things will lead to a huge advantage on the field of battle.
However if you build to a point where army A and army B are 100% equal in everything but army B has better weapons. Then, and only then, can you always expect army B to win.
So it's all about context and what you do with the story and how you set up the two conflicts.
Personally I think that capturing and trading equipment could help your bone and wood army. At least give the cavalry bronze headed weapons.
More numbers is also an obvious point. So society will be impacted.
You can also have "castles" but that will require special knowledge but it's your story.
So you can go far with that.
TL:DR it's all about context and the actual general in command of the army.