In Black Ops II zombies, what are the differences between public match and custom games?

In Black Ops II zombies, what are the differences between public match and custom games?

The main difference that I know is that in custom match, randoms cannot search your game as in public match, but friends can join a game in progress.

Here are the descriptions in the zombies multiplayer menu:

• Public match:

  Find and team up with fellow survivors to fight against unending waves of the undead, post to leaderboards, and review slaying stats.

• Custom games:

  You make the rules! Play with fellow survivors and team up to fight unending waves of the undead.

Do custom games have the same leaderboards as public match? The custom games description excludes mentioning leaderboards and stats.