The... shape?

This might be a silly answer but that is one advantage in some cases where you actually need a shape.

When you are putting the flash inside a softbox there is no advantage, but it might start to have one when you are using different attachments to the flash.

One example is using umbrellas. A lot of users do not realize that to maximize the diffuse power of an umbrella covering all of it and not spilling light is a really important thing.

Compare how the light hits (ideally) this umbrella

This can be also the case with some modifiers like a parabolic head. (But it is funny how people do not pay attention to actually put the light on the focal point of the parabole)

This is also true if you actually want a round projection, for example in a wall.

A round shape can be less distracting than any other shape.

One selling point of the round head flashes is that they have a more controlled fall-off. A quote from one webpage

with a soft smooth fall-off

But, on the other hand, normally the initial premise for a rectangle head is to cover the same as the frame (red) when the flash is pointing right at the subject with an on-camera flash.

If you are making some cookies, for example, to shoot thru some Venetian blinds, probably a rectangle shape will be more useful than a round one.

But in the end, it is just a shape.