What exactly changes when selecting either of these options on a modern tv set that works fine on both settings? I tried both and couldn't tell the difference.
Are there any films/sources that are better viewed in one over the other?
Your TV can be 50 Hz or 60Hz, that is 50 or 60 frames per second. It's possible that it can work both as a 50Hz and a 60Hz TV. It is possible (even likely) that it will convert the wrong input to what it can display, by adding extra frames or removing unused frames. That will reduce the quality because for example displaying 50Hz input on a 60Hz TV means every fifth frame is displayed twice.
Your video material is whatever framerate it is. You can't change it. Your Apple TV will convert it if necessary to the framerate that you said your TV is - either 50 or 60 Hz. Video material at 30 or 60 frames per second can be nicely converted to 60. Video material at 25 or 50 frames per second can be nicely converted to 50.
You will get the best possible results except in one case: If your video material matches well with the TV (for example 30 fps video, 60 fps TV) and you set your Apple TV to 50 fps, then first the 30fps video is converted to 50fps which isn't good, and then the 50fps is converted to 60fps, which isn't good either.
If your video material doesn't match the TV (30 or 60fps video, 50fps TV) then it doesn't matter much how you set the Apple TV and it won't make any difference, because either the Apple TV or your TV has to make a conversion. Your setting only affects which one makes the conversion, the results should be indistinguishable.
In any case, you'd have to look hard to see a difference. Still best to find out what your TV actually supports and set the Apple TV to that. Typically European DVDs and TVs will be 25/50 Hz, US DVDs/TVs will be 30/60Hz.