I don't understand why I'm reading that after a hard fork, the existing transactions would be on both chains but the new transactions will only be on one or the other (resulting in 2 different chains, thus an altcoin). Won't even new transactions eventually end up on both chains?

Transactions don't care about block size, so both versions should accept them. Even if one chain has already included it, software for the other chain won't care about the first chain, so it should still have those transactions in its pool, resulting in it being put into its chain as well.

I don't see how this will result in 2 separate chains... it seems like both chains will always have approximately the same exact transactions, just possibly in a different order and grouped into different blocks.

The only exception to this I can think of is if a double-spend is attempted. Then it's possible that one of the chains will accept one of the double-spend transactions and the other chain will accept the other.

But otherwise, I don't see how there would be a difference.