There are several reasons I can think of off the top of my head:
Geography, Tradition, Economic Reasons, Religious Reasons
Let's analyze the Vesprians. Why are bows traditionally used?
- Long range
- Cheap/easy to manufacture if certain wood is available
- Potentially very powerful (a skilled longbowman can take down a man in plate armor at 100 - 600 m)
- Versatile hunting tool / weapon of war
So let's make up a bit of backstory for them. Let's assume that theirs is, perhaps, a historically a kingdom not very rich in metals. Let's further assume that a large portion of their territory consists of woodlands, hills, maybe even rocky mountainous terrain where traditional military formations can't easily maneuver, etc.
However, a troop of rangers hiding in the tree tops can unleash hell on an enemy unit trying to advance through their territory.
Thus, their reason for using bows can stem from the geography of their territory, and become tradition in time. Historical example? The British longbowmen which have served them well in war, and become near legend.
Now let's take a look at the Gremians. Horsemen, you say? That's easy.
Have their territory consist of sweeping, open plains on which vast herds of wild horses roam free. Gremians have a long tradition of taming these horses, and, over time, have developed very effective riding, fighting, and military techniques, all centered around their biggest advantage: their unmatched horses.
Those vast plains mof theirs also make for perfect terrain for Shekarian military formations and tactics, making a pitched battle suicidal for the lighter-armed Gremians. Thus, they instead focus their efforts on developing very mobile mounted units which can't get pinned down and engaged in a pitched battle, and which can more easily flank enemy infantry.
They live underground - animals don't do well there (even in their vast halls), so they never developed an affinity for riding any sort of beast. Instead, they put their stocky, powerful frames to good use as heavily armed and armored infantry, which works very well when engaged in pitched battle in some dark, narrow tunnel.
Dwarfs are unmatched craftsmen, and build massive underground enclaves for their clans, and families. These are typically designed with defense against the many terrors of the deep, and other quarrelsome clans in mind, and have very heavily defended entrances, not to mention thick walls of solid stone.
As such, dwarfs have put their natural tinkering talents to use in inventing all sorts of nifty gadgets to break through the thick walls, and gates of their enemies. In fact, they've developed some truly impressive siege engines which they carry to the battlefield in pieces, to be assembled in the enemy's main hall, and used to turn the various structures within to rubble.
When dwarfs eventually turned their attention to the riches of the surface, they found that those same weapons of war work even better when you don't have to worry about bringing the ceiling down on your head, and they enthusiastically designed even more powerful, and deadly contraptions.
Becoming a mage is a difficult proposition. It involves a lifetime of dedication to the magical arts, and very few are able to gain even the basic knowledge which will get them started down this road. Furthermore, there seems to exist a certain ... magical ability which some rare few are born with, and which allows them to progress down the Path of Power at nearly unimaginable speeds. People to whom magic seems to come naturally, and whom, especially when properly guided and trained, become mages of great, and terrible power.
Nobles and other powerful lords have often sought to gain this knowledge for themselves, feeling that having mere commoners gain such abilities is upsetting to the balance of power in the land (aka, they are competition). Historians can list many noble houses which have tried to breed this power into their ranks, and nobles who have expended vast amounts of gold, and other riches to have a mage instruct them in the ways of magic. However, few of these efforts have ever born more than a modicum of success, almost as if the Gods themselves sought to keep magical ability, and prowess out of the exclusive grasp of the elites.
Angered by their inability to gain this power for themselves, noble houses have long sought to have magic banned, or severely regulated, and many bloody conflicts have come to pass over this issue.
More recently, however, some reckless, if well meaning, mages have accidentally unleashed destructive powers which ravaged the lands, and gave their enemies exactly the excuse they needed to turn public, and political opinion against them. Hunted down and imprisoned, if not outright executed, mages fled to a land where they could regroup, and band together to impose their own laws, and rule, without fear of interference, or harm.
Roman troops were professional, dedicated, and well trained soldiers. Their choice of armor, and weapons carefully went hand in hand with their tactics and strategy. You can explain it much the same way (read up on how effective Roman tactics and formations were, and how carefully crafted/chosen their equipment)
An easy explanation for them not widely using cavalry is that their nation doesn't have a lot of horses available, and perhaps there even exists a social dislike for the animals (religious perhaps?).
Bows? Maybe these guys are very big on martial prowess, and dislike "cowerdly weapons".
As you can see, there are plenty of reasons for the world to be shaped the way you want :)