I had a little time to do some testing in relation to my original post. If you are interested in this topic, read on ...
I used two albums for my test data. Album A contains 17 songs by 6 different artists. I set Part Of Compilation to Yes for all 17 songs. Album B has 25 songs by 7 different artists. I set Part Of Compilation to No for all 25 songs.
I sync-ed the two albums to four iDevices: an iPad 3, and iPhone 4S, an iPod Touch (4th generation) and an iPod Nano (5th generation). I sync-ed each iDevice from my iTunes library on my Mac. For each iDevice, I selected the two albums from the Albums section within the Music menu. Album A appeared as one album in the Albums section, so I had to select only one tick box. Album B appeared as 7 different selections, so I had to select 7 tick boxes in order to get all 25 songs to sync to the iDevice.
So here is one difference using Part Of Compilation: If you sync individual albums to your iDevice, you will have to remember to select all the tick boxes pertaining to the songs on the album (7 in this example) when Part Of Compilation is set to No. If you simply sync all your music to your iDevice, or if you sync by Playlist (for example), this will not be an issue.
After sync-ing, I tried to play each album on each iDevice. I opened the Music app and selected the Albums option at the bottom of the screen in order to see songs by album. Album A appeared one time in the album list. I was able to select it and listen to all 17 songs. Album B, however, appeared 7 times in the list. I selected the first appearance of the album in the album list, which happened to contain 3 of the 25 songs (the 3 songs were all by the same artist). The iDevice played these 3 songs then stopped.
The above playback experience was consistent on all 4 iDevices that I tested.
So here is the next difference using Part Of Compilation: If you want to play an entire album, you need to set Part Of Compilation to Yes. If it is set to No, you will get multiple entries in the album list, each containing a fragment of the whole album (one fragment for each artist). I did not see a way to play the entire album (unless you create a Playlist, of course, containing all songs for the album).
If you have a different experience from my testing, pls let me know!
Guys, this might be helpful too... I have an iPod Classic 160GB latest generation. I realized I had a couple of aritist that supposedly were synced to my iPod (at least that's what iTunes said when plugged in), but when the iPod was unplugged, it wouldn't show up in any of the menus in my iPod. I even tried doing a manual search in the menu Music/Search, and nothing; like if they didn't exist. Anyways, I realized the option "Part Of A Compilation" was chosen on those that were not showing up, so just for the sake of my own peace of mind, I unselected this, and resynced my iPod, and voilà! After reading your post, I realized you tried several iDevices, but didn't try the iPod Classic, so I thought I should post this too for the other guys out there that might have this issue as well. It seems like the "Part of a Compilation" option works great for those other iDevices like the iPod touch, iPads, iPod nano, but not on the iPod classic. Hope this helps!
Guys, this might be helpful too... I have an iPod Classic 160GB latest generation. I realized I had a couple of aritist that supposedly were synced to my iPod (at least that's what iTunes said when plugged in), but when the iPod was unplugged, it wouldn't show up in any of the menus in my iPod.
On the iPod, Settings > Compilations to On.
Songs marked Part of Compilation will then be under Compilations.