I'm unsure what carries over from game to game. I thought the defence grid bonus did as it says it is permanent, but after getting it to 30% before losing the game it started again at 15% on the next play through.
The only thing I have found so far is that you can keep one pilot from the game and take them into the next game with their XP/level in tact. Also once pilots are unlocked, you can choose which one to start with, but they start with zero XP.
Coins from achievements are also permanent, but they are only used for unlocking new Mechs, or is there something else they can be used for?