Bitcoin public key — why do we have a compressed and uncompressed version?

My understanding is that the public key can start with 04 if it’s uncompressed, or 02(x-coor) or 03(y-coor) if it’s compressed.

What’s the purpose of this?

Greater entropy as we can have a larger public key if it’s uncompressed?

And if this is indeed the case, why would you want a compressed public key?

Wouldn’t it be safer to use the larger one?