I've read completely through a couple of guides and tried many things out for myself. Here is what I have learned:
There is really nothing that changes for your character between games. The only one that does is that you lose any spell effects that you currently have, which you would anyway for simply entering town.
- Modifications to your max mana via Sacred, Ornate, and Fascinating shrines persist to the new game
- Modifications to your max health via Black Death zombies persist
- Items you are wearing and carrying persist. Items left on the ground in town do not.
- Spells effects do not persist such as Infravision and Mana Shield.
- Any town portal you have does not persist.
- Openings to Caves, Catacombs, and Hell are based on your level in multiplayer and therefore appear in the new game. In single player, these openings are always closed on a new game.
Not all item stats persist to a new game!
- The Lightforge mace morphs when you carry it to a new game. (I haven't seen this happen and have no idea what it morphs into)
- Effects from the Weird shrine do not persist to a new game. This was doubted (for good reason), so here is some proof:
Sword affected by Weird Shrine:
Same sword after starting new game:
- Effects from the Gloomy shrine do not persist to a new game. This was also doubted, so here is more proof:
Helm affected by Gloomy Shrine:
Same Helm after starting new game:
- Effects from Hidden Shrine do persist. The helm pictured above was also affected by a Hidden Shrine.
- Changes based on Warrior's Repair skill and Sorcerer's Recharge skill persist.
- Current durability and spell charges persist
Note: Any effect received from a Goat Shrines or Cauldron will persist in the same manner as one from the named shrine.