That depends on how large the half spheres are :)

Since we're all living on a sphere, at some point the difference is moot. I've been in a couple of dome houses and we had a museum here in NL that was built inside a geodesic dome (Aviodome, near Schiphol, since dismantled and moved to Amsterdam:

https://nl.wikipedia.org/wiki/Aviodome

and

https://amsterdome.nl/

What I love about the story is that the building was simply dismantled and put up again in a new location.

That’s not a half sphere. Pure half sphere structures are extremely rare as they waste a lot of space and building materials. Your better off using more than half a sphere so the walls curve in slightly at the base. You get more useable internal volume from a slightly smaller exterior footprint and use less building materials. The next obvious improvement is to use a dome on top of a cylinder which again improves useable space over a half sphere and is a fairly common design as seen with silos and observatories etc.

The only way a half sphere works out well is if your dealing with some form of pressure dome. Either using internal pressure to support weight, or in the case of a building underwater/Moon/Mars dealing with very high pressure differences.

(Geodesic) domes are not usually half spheres at all. They can be but they really don't have to be. You can do more than half a sphere, half a sphere, less than half a sphere and all of them would be called domes.

https://en.wikipedia.org/wiki/Geodesic_dome

Some good sample images there.

And a geodesic dome does not require any internal supports, so it doesn't matter whether a pressure differential is involved.

Essentially any slice of a sphere can be made into a geodesic dome.

Geodesics usually have flat panels as well - flat panels of limited size meeting at odd angles, yes, but flat panels nonetheless - that allow you to use conventional furnishings against or nearby to walls without unusual visual gaps in at least a few places. "Real" domes, on the other hand, not so much.

As the frequency goes up it starts to approximate that dome better and better.

Must be. There have been some examples of people in Scandinavia building a house inside a dome!

The dome provides a greenhouse enviro and then, inside that, they build regular old rectangular houses: https://www.loveproperty.com/gallerylist/66379/this-family-l...

Those are beautiful.