A stronger version: you can't even tell the real shared secret apart from a random point.
The idea:CDH says an eavesdropper can't compute the shared secret. DDH says something stronger: shown a candidate, they can't even tell whether it's the real shared secret or a random point. That matters when the shared secret is used as a mask: if any pattern showed through, an attacker could learn part of the message, such as whether a vote was yes or no.
Why it matters: Needed for encryption that hides every last bit of the message.
The details
On the map: on the curve
Given aP, bP and a point R, decide whether R = abP or a random point. A CDH solver also solves this, so it is a stronger assumption[1].
It is what makes ElGamal encryption hide everything about the message[2], and it underlies DDH-based oblivious transfer[3].
This assumption gives key exchange on its own, so it needs more than one-way functions and falls if public-key encryption is impossible.
TD
This assumption gives trapdoor functions on its own (computable by anyone, invertible only with a secret), so it falls if trapdoor functions don't exist. Schemes built on it need not use a trapdoor: ElGamal doesn't.
References
Dan Boneh. The Decision Diffie-Hellman problem. Algorithmic Number Theory, 1998. doi:10.1007/bfb0054851
Taher ElGamal. A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms. Advances in Cryptology, 1985. doi:10.1007/3-540-39568-7_2
Mihir Bellare and Silvio Micali. Non-Interactive Oblivious Transfer and Applications. Advances in Cryptology — CRYPTO’ 89 Proceedings, 1989. doi:10.1007/0-387-34805-0_48