I'm about to move a spider and I usually use the cup/glass way, but that would seem pretty horrifying if I were the spider. I'd prefer to use my hands, but they don't seem to like the texture of human skin. Why? I've googled and not found anything beyond "they don't like people", which I get.
According to arachnoboards:
I'm guessing that it's that they don't generally prefer mushy surfaces that are soft and pliable, because they are less supportive and more likely to give way, causing a potential fall. Falls are bad news for them.
...including not only tarantulas but also amblypygi, phasmids, beetles, mantises, roaches, millipedes, etc. that they don't seem to like abrupt changes to the surface they are walking on. It's like they're comfortable with what they're on, and don't want to switch to something new - whether that "something" is my hand or arm, or a catch cup, or a piece of paper or egg crate or bark. Once they have been on the new surface (such as my hand) for a few minutes and gotten acclimated, they are equally reluctant to get off it - even when that means returning to their former substrate.