iTunes does not copy photos to your iPad. It syncs a selected folder of pictures to your iPad. The difference is that syncing means when you change stuff on the computer it changes the contents on the iPad, the next time you connect. When your removed the one folder and added a different one you were not telling iTunes to copy the new folder to the iPad. You were telling it to stop syncing the first folder and sync the second one instead.
To get around this you copy all the folders you want to sync to the iPad into one master folder, then select that folder in iTunes Photo sync settings. Each folder in main folder will become a separate album in Photos on the iPad.
Only one level of sub-folders is recognized. A folder inside a folder inside the main folder will be ignored.
If you choose the Selected Albums option you can turn on/off folders within the main folder. Since you already have a folder with your album folders in it, and only want some of them, this would probably be the best option for you.
On the other hand, if you have the hard drive space, making a separate folder just for you iPad albums and placing 'copies' of the folders you want there would give you more flexibility. You could move, copy, delete pictures without worrying about deleting or altering your originals.
iTunes: Syncing photos
Hold in mind that albums synced to the iPad using iTunes must be removed the same way. There is no way to delete these albums directly from the iPad, short of restoring the device.
None of this has anything to do with iCloud and Photo Stream, which is a newer service.
iCloud: My Photo Stream FAQ
iCloud: Shared Photo Streams FAQ