I don't know mucho about Cryptography, I try to understand, It's for homework.
Why the standard DH is not sufficient to ensure the security of the Diffie-Hellman key exchange?
My answer was:
In the standard Diffie-Hellman key exchange protocol, both parties compute the secret value gαβ for two privately chosen values $\alpha$ (by Alice) and $\beta$ (by Bob). An eavesdropper can publicly see the pair ($g^{\alpha},g^{\beta}$). So, if checking whether, for some $g^{\gamma}$, it is the case that $\gamma = \alpha \beta$, were computationally easy —i.e., the DDH assumption fails—, then the eavesdropper would have found $g^{\gamma} = g^{\alpha\beta}$ without disclosing $\alpha$ nor $\beta$ explicitly, even if the DH assumption is true. This is why the CDH/DDH are the assumptions that really matter in the key exchange.
