I think Liam Morris's answer about Steam-Powered Weaponry is a really good starting place, but I feel it does not fully represent the more specific question of how weapons would have advanced. Original gunpowder cannons, like the original Steam-Powered cannons, were garbage. They were in many ways worse than the catapults and ballistas of their time, but both weapons saw some niche uses, until people figured out how to make their reliability and rate of fire better.
Instead of the steam-cannon as the solution, I think it should be viewed more as mankind's early steps into the realm of compressed gas weapons. In the late 1500s, when firearms were just starting to dominate the battlefield, we also started to see the advent of compressed air guns. These early air guns were just as powerful as the muskets of their day, but were much more expensive to make. Without gunpowder, these firearms would have dominated the battlefield compared to bows and crossbows, but like a knight's plate mail; would have been too few in number to completely phase out other weapons the way muskets did.
In the late 1700's, Tyrolian invented the Girandoni air rifle which was a semi-automatic 20-shot air rifle. The same way we saw the revolver cylinder push firearms into being the undisputed best weapon, this same innovation would be largely responsible for air rifles completely phasing out other ranged weapons.
By the early-to-mid 1800s, industrialization would be adequate that compresses air guns could start being mass-produced as weapons for the common soldier. By the time of the US civil war, we'd probably start to see armies equipped entirely with air rifles.
Eventually firearms would go toward prefilled compressed gas tanks much like a paintball gun. While paintballs are super squishy non-lethal blobs fired under 700-800 psi, the same technology can easily be used to fire very lethal lead slugs under 3500-6000 psi giving you a weapon that is in every way just as lethal as a modern assault-rifle. The only major drawback being heavier ammo due to those bulky air tanks, and larger slugs to compensate for lower muzzle velocities. But if bullets are not an option, it's still a small compromise when considering the utility advantages vs any medieval weapon alternatives.
But it would not stop at riffles. Artillery would also find its counterpoint in the form of Light-Gas guns. While Light-Gas guns of today are not designed for easy reloading, that is just a matter of necessity not pushing along invention. Creating shells that are integrated with their own rupture plates would be a trivial matter of engineering meaning you could make artillery capable of firing meteor speed projectiles in a simi-rapid fashion that could turn castle walls to dust, sink steel-hulled battleships, and do all the same things we do with modern cannons, and probably better.
Stirling engines would probably be the best option to replace internal combustion engines for use in unarmored vehicles, but their weight and inefficiency at larger scales would make tanks and planes much harder. In WWI, internal combustion engines were barely efficient enough to make the British Mark Series tanks even move to the point that they faced frequent issues with total engine failures which would only be exasperated by the even heavier stirling engines. So, tanks would be slower, less armored, and less well-armed, but probably still achievable. The need for better heat dissipation would also also force engineers to leave the engine section unarmored and mostly exposed. Planes would suffer a similar handicap since limiting weight is such a vital aspect of achieving flight. They would also lose a lot of their utility without any heavy explosives as an option for bombing, but as scouts and transports, they'd probably still see their fair share of use. When modern electric engines come out, they could probably make airplanes and tanks of reasonable quality though. Because of the advantages of water cooling, stirling engines are already often used as an alternative to combustion engines for powering large warships; so, the arrival of advanced naval mechanization would be much less affected.
The only area of modern warfare I do not foresee an equivalent technology developing for without combustion is missiles. Chemical fuel is the only modern technology that I can think of that would offer the thrust-per-mass to make self-propelled projectiles nearly as effective as they are today. It is also possible we would not have achieved space flight yet meaning no GPS, less globalized communications, etc... unless someone could actually make a working nuclear pulse engine.
As for Armor:
Early air rifles fired higher mass lower velocity slugs than muskets to deliver similar stopping power. They may not penetrate heavy plate armor the way a musket or bodkin arrow did, but similar to a Roman sling, it's heavier slugs could have transferred enough kinetic energy through armor to cause fatal internal hemorrhaging anyway. Plate mail over thick padded armor may have stuck around longer than in our world if it could distribute the force well enough without being penetrated, but chainmail, scalemail, brigandine, etc would have all become obsolete at about the same time.
Shields may have lasted longer too. Because a shield does not rest against your body, non-penetrating shots could be better stopped without causing massive internal injuries.
Modern body armor would see some minor, but meaningful changes. Modern armor is designed to stop something like a 7.6×7.6×28mm slug fired at ~60,000 psi. In contrast, an air rifle with a 6000 psi tank would need to fire an 18.7mm round-rifled slug to achieve similar stopping power. The air rifle round would hit slower while being ~4 times as massive as aforementioned slug. This means that armor could use thinner exterior plating to achieve the same impact distribution for impact distribution and thicker interior composites to absorb more energy. Because exterior plates are normally the heavy bits, body armor of a given weight would be much more effective. This could force combat into much shorter ranges where you may need to close within a few feet of a well armored combatant to achieve a fatal shot.
Tanks and ships would face the opposite concern. Light-gas guns have tremendous muzzle velocities; so, they would probably adopt a double hull spaced armor system similar to the ISS in order to protect against light-gas guns. The outer hull being designed to ablate and distentigrate the shot, while the inner hull blocks the plume of smaller debris after it has spread out some.
A Final Thought:
It's hard to say just how early or advanced any of these innovations would actually be. In our world, alternative solutions have always been secondary to combustion technology, but if all the research that went into our version of guns, engines, etc instead went into other systems, we might have started fielding things like light-gas guns, railguns, and/or laser weapons much earlier out of necessity. So what we would have today could very well look like something out science fiction.