What Rowan said above, plus ...
We have a variety of touring styles. We like point-to-point tours of various lengths, hub-and-spoke tours, multi-mode tours, etc. etc.
On our current RTW tour, we've gone with quite a mix of touring styles.
For example, we flew into Taipei, then took a shuttle to the train station, then arranged to have our bicycles put on the train, then took the train to Hualien. There, we did a 4-day hub-and-spoke tour before repeating the journey in reverse back to Taipei. We were able to do that with our full-sized touring bicycles, but it was a hassle.
-- It would have been so much easier to transport folding bicycles in the shuttles. On the way to the train station, we had to arrange a shuttle that did not have any other passengers, so our travel time was at the mercy of the shuttle availability. On the way back, the regularly scheduled shuttles were full and we had to pay for a special shuttle. If we had folders, we could have put them on any shuttle and they would have saved us money.
-- On the train, it might have been possible to put the folders in the luggage area rather than having to make special arrangements for them in an unfamiliar language.
-- The full-sized touring bicycles fit in the hotel we stayed in all right, but the folders would have been easier to transport up and down, and would have left us with a bit more elbow room.
Another example, when we got to Hokkaido, Japan we found out that full-sized bicycles are not allowed on the trains. We had thought we might cycle northward or eastward from Chitose, and then catch a train back when our 10 days was up, but since we couldn't use the trains, that idea was out, so we had to come up with another plan. We'd like to have the option of catching the train on our next trip there, and I believe folding bikes are allowed on the trains.
Folders could also replace the hassle of rushing to put the bicycles onto the bicycle car of the trains in Europe, hoping there would be enough room for them, paying extra for them, having to carry the bicycles up and down stairs when the lifts weren't there or were out of order or were too small, etc. etc. And ... I probably wouldn't have fallen out of one of the trains when we encountered a difficult bicycle storage situation (see the Switzerland part of my journal) if we had folding bicycles.