I can make a separate write up on showing a comparison of the two in a different post.
The short answer is that the MMD approach isnt a full a State proof. The Relayer is not posting a Patricia Merkle Trie(Current State model of Parachains) membership proof of the source chain in which the receiver checks against a latest state root. Although technically with this design you could do full state proofs if you really wanted to because the BEEFY root can verify any parachain header.
So while MMDS do use Merkle co-paths as proofs it is not a Merkle co-path of the State tree of the source chain. I hope this helps.