Is a seed + passphrase really that different from just the seed?

Having read plenty of discussion about passphrases, this is what I understand - adding a passphrase to your seed actually generates a whole new wallet address, it's not like just adding an extra layer of security to your existing wallet and seed.

Given that it seems to be generally agreed that 12 word seeds are enough and 24 is overkill, then why are 13 words necessary? Really then, the only difference is that your 13th word is not meant to be written anywhere, only saved in your memory. So, actually, just memorising the last word in your 12 word seed and only storing the other 11 words would be just as secure, the only difference there would be whether you trust the initial presentation of the seed, that is, your 12 word seed was once shown on your computer screen which was probably connected to the internet, whereas a passphrase has only appeared in your mind. But given that most people seem happy with the level of security involved in your seed being created and shown to your on your device, I wouldn't think that's too much of a concern.

The last point to consider, is that memories can fail, or people can die and so memorising your passphrase can be risky, and because of this, many people decide to also store their passphrase , albeit in a different location from their seed phrase. So, in that situation, it's really no different from splitting up your seed and storing parts of it in different locations. Correct me if I have any of that wrong.