I'll start with some background in the centuries leading up to late antiquity, so bear with me.

Round/oval shields are more versatile and easier to produce than earlier scuta. The squared scutum ("legionary's shield") had very specific applications and was from a time when close-order infantry combat dominated warfare in the Mediterranean and skirmishing was common in northern Europe. The pronounced corners, rounded or angled, allowed legionaries to form a solid wall that could deflect missiles from hitting the (comparatively vulnerable) feet and other extremities, and the horizontal curve of the shield offered some protection from missiles and utility in deflecting pointed weapons, which would be shunted away from the body of the user in, for example, a charge against a spear- or pike-armed enemy. Were I to go out on a limb, I would argue that this was a significant contributor to Rome's victories over its opponents (especially the spear- or pike-dominant armies of the Greek states and Carthage) during the Republican period and into the Principate. (I can't speak to the Samnite Wars, but the Romans found Samnite equipment and tactics revolutionary and adopted many of them. I don't think the scutum originated there, but I've reached the limits of my knowledge on the topic.)

There's a lot of discussion regarding the use of the shield as a "punching" weapon (using the boss as an analogue for brass knuckles), but I don't find that a persuasive motivation for the use of the scutum over other models. [EDIT: I also don't view the scutum as being an item that would lead, through its use, to the development of "punching" tactics.] The scutum was not the only shield to have a centrally mounted metal boss, and the dimensions of the shield would make such punching no less awkward than it would be with a contemporary bossed shield of a different shape. EDIT: I've done some further research, and it appears that the boss of the scutum was sturdier than other shield bosses of the period, lending credence to the "punching" tactic.

The shield was not without disadvantages: Axe, falcata, or falx blades (which were effective in delivering a strong cutting blow against a small point in the shield) regularly cut through the scutum. The flat top and bottom edges did nothing to deflect these weapons, in contrast to the natural deflective benefits of a round shield. The scutum was made of a thin composite to reduce its weight; the thinness of the shield was also problematic in these instances.

As the Romans reached the frontiers of their world, they began facing opponents who were almost exclusively versed in very different styles of combat, in which the earlier scutum was less effective. Outside of open ground, the scutum presumably would catch on brush and undergrowth and would be difficult to operate in, say, a forest in the German borders. And useful though the shield was in deflecting missiles, a (comparative) lack of sturdiness in the shield's construction would reduce its utility in repulsing a charge from armored cavalry. (There's a concomitant re-adoption of spears over the more lethal but less versatile pilum [Roman javelin] during the period when Rome's and Iran's borders began to encroach upon each other.)

This led to the re-adoption of the round clipeus, which was easier to use and manufacture and had a greater degree of durability and utility. It could be used by mounted troops or infantry; just as crucial to your question is the importance of size (not simply shape), because larger shields are harder to use on horseback than smaller ones. Another advantage is that though the scutum provided better individual protection from blows or missiles, the overlap of round clipei could produce a stronger wall of shields to stop charges from both infantry and cavalry. Regarding tactics, the Republican mode of infantry engagement often began with the throwing of pila followed by a massed charge, in which individual protection was more important than the creation/maintenance of a shield wall. A scutum works very well here, but it works less well when a unit is preparing to weather a charge.

To circle back (hah!) to your question, both round and oval-shaped shields were used likely without thorough standardization throughout the ranks. I forget the specific point when this happened, but during the fifth(?) century, the provision of armor to soldiers was replaced by a cash stipend, which the soldiers were expected to spend on their own equipment. It's been posited that many chose to pocket the money, though my only source for this is a comment from the later Strategikon about the importance of putting the best-armored troops in the first ranks of a formation. (I HATE to be so vague about this, but I don't have sources handy.)

It's important to bear in mind that even with the "mass production" of weapons and armor, there was no uniform specification to which those items were made; this means that any two shields would not be identical copies, as would be the case with modern industrial methods of manufacture.

So in short, both were used, though it appears that circular shields became more common over time. Illustrations are naturally suspect (cf. the use of the lorica segmentata on Trajan's Column).

I'd be happy to discuss this further! Apologies in advance for the meandering and thrown-together nature of this answer.