I'd like to hear if there's a real difference in sound between two similar builds, one with and one without.
The sound you hear from any of this family of instruments is created by several linked components. First the string vibrates the bridge. This vibration is spread to the top by the saddle. The vibration is now a sound wave that pushes out from the top. It also pushes inside where it bounces around before coming out (through the soundhole and partly back through the top as well, and minimally through the sides and back). The sound that reaches a listener is actually a complex combination of waves.
The size of the top, size of the inside space, size of the soundhole, thickness of the wood, bracing, type of wood, size of saddle, bridge material, type of back and side material, tension of the strings - all of them and more play a part in making the sound your listeners hear (and that you hear - but more on that below).
So the general answer is: yes. A cutaway
will affect the sound simply because it alters at least two of these components. It makes the top surface and internal space smaller. But the
real question is: how much will it affect it?
Larger spaces and surfaces are able to create/transmit lower frequencies better than smaller spaces. So the loss will be partly in the lower end of the range, partly in the complexity of the harmonics and somewhat in the amount of volume produced. This can be compensated for by creating a larger space internally and making the lower bout wider and maybe even increasing the soundhole size.
A simple demonstration of the difference size makes is to play a tenor beside a similarly-shaped soprano. You can hear the greater volume and tonal complexities in the tenor.
But in realistic terms, the effect of the cutaway will be small to negligible for several reasons. First, the uke is already a small body and that limits its sound output right from the start. Second, its tonal spectrum tends to the higher end anyway, so there aren't as many low-end frequencis to lose (I haven't tested it yet, but I suspect it could be measured more noticeably with a low-G uke than with a high-G). And finally, 90-odd percent of the sound made is projected forward, not up, so you won't likely notice anything as a player.
And of course if you're amplifying the sound you can compensate for and change any original output, so then it becomes more an aesthetic choice than an acoustic one.
I have ukes with both designs. I like the look of the cutaway but it has little to no impact on my playing. The frets in the cutaway region are generally too high up the scale and too close together to get much of my attention, and are easily reachable anyway on any standard uke, so the cutaway offers minimal advantages for playability. But it looks cool.
Therefore the answer is: buy the uke(s) and designs that appeal to you.