When I double-click on pycharm.sh, Ubuntu lets me choose between "Run in Terminal" and "Run". What is the difference between these options?

To make it a bit more user-friendly: After you've unpacked it, go into the directory, and run bin/pycharm.sh. Once it opens, it either offers you to create a desktop entry, or if it doesn't, you can ask it to do so by going to the Tools menu and selecting Create Desktop Entry...

Then close PyCharm, and in the future you can just click on the created menu entry. (or copy it onto your Desktop)

To answer the specifics between Run and Run in Terminal: It's essentially the same, but "Run in Terminal" actually opens a terminal window first and shows you console output of the program. Chances are you don't want that :)

(Unless you are trying to debug an application, you usually do not need to see the output of it.)