Champs and Items should be the same on ALL maps.
It's a straightforward game design concept: champs and items should be the same on all maps.
For example: I chose Singed in Hexakill once this week, then when I played him it was strange --- something was different from normal about this champion! Some googling revealed that Singed has received nerfs in the past specifically for the Twisted Treeline map, so when you play Singed on Twisted Treeline he is not as strong as normal Singed. There are some big problems with this, the most obvious being transparency. If you are going to have "adjustments" to the champions depending on what map they are on, those adjustments should be CLEARLY AND UNAMBIGUOUSLY MADE VISIBLE to the player selecting the champion.
**The information that I was able to find about Singed nerfs on TT was little bits and pieces buried on the LoL wikia site, not anything comprehensive or authoritative, so for all I know there is more than what I found, and there are presumably other 'adjustments' to other champions for different maps that are by no means transparent. **
When you have a supposedly competitive game where players are spending many hours trying to master different champions and game dynamics it seems like an obvious principle that those champions should play the same regardless of what map they are on.
And don't even get me started about all the special Twisted Treeline items and the lack on that map of many standard items... If you have to remove certain commonly used items (such as Rabadon's Cap) and add other similar ones depending on what the particular map is, then you need to seriously take a look at your itemization overall and figure out why it is broken. Btw, this situation with the items is extremely confusing for a new player (which I recently was). Nothing in the item list builder tells you which items are available on which maps, much less why.