The really really short answer is: money and knights.

This is still pulling out a few key factors from a complex history, but here's a longer explanation.

The Roman Army at its peak was built around a disciplined, well-trained and uniformly equipped core of professional soldiers. The problems of command and control on an ancient or medieval battlefield were tremendous. The training, structure and professionalism of the Roman Army overcame some of these problems, so commanders were more able rely on their troops to carry out their orders. These factors allowed the Roman Army to use relatively sophisticated tactics.

However this sort of force was expensive to maintain. You're constantly paying professional troops, you have to reward soldiers and officers with land, pensions, booty... Keeping the loyalty of the legions was a big problem. As the Roman Empire declined, so did the financial and organisational resources of Western Europe. Up until the 15th century, nobody had the resources to maintain a force like the Roman legions.

So what were armies like in the dark ages? To start with, they were smaller. They were more likely to be based around the retainers (or bodyguards) of nobles (eg. Saxon housecarls). These people were given the money and the time to prepare for war. These forces might be supplemented by levies of the populace (eg. the Anglo-Saxon fyrd). These forces were inexperienced, harder to control and less well equipped.

To continue looking at the Anglo-Saxons, although they had round shields and axes rather than rectangular or oblong shields and swords, the shieldwall was still their standard tactic. And it could be a very effective one, particularly on the defensive, where it's a reassuring and natural formation. Its last famous use was at the Battle of Hastings where it acquitted itself well, only failing the formation broke up.

Which brings us on to the subject of cavalry. The Norman army contained both cavalry and archers, but it's the latter that are more important (except in England and Wales after the spread of the longbow). The dark ages saw the spread of important technologies: stirrups, higher-backed saddles and larger breeds of horses. Riders could control their horses much better with both hands occupied, they could hit harder without falling off their horse, and the horses could carry more heavily armed and armoured men. This was accompanied by socio-economic innovation - the creation of a class of men equipped and trained as heavy cavalry - the people we usually call knights.

So it was the knight that was the most important element of most medieval armies. Well trained, well armoured, fast moving, hard hitting. Although in theory an infantry formation that stood its ground could resist a charge (a horse won't charge a solid body of men - it's not stupid), in practice the momentum and terror of a charge was usually enough to open gaps or make a formation break, and once that happened the cavalry had the upper hand. With some notable exceptions the knight was the dominant force on the medieval battlefield until the disciplined pikemen and handguns became common.

Speaking of exceptions, England in the mid-late Middle Ages was one, as knights tended to dismount and fight on foot, between wings made of longbowmen. Scotland was another - as a relatively poor kingdom, it was less able to raise large forces of knights and relied more on infantry.